1^3 

37 


UC-NRLF 


TUDY,ft 


>:^;^>;t>M-;>s^W^;-\^^i«'^^k;^\-s 


S5JSS5sS\      J 


^S«11^S 


a\  «« 


^cS>iS^.«;.^~;\S;,<.>;^ 


ii^v 


i* 


'**: 


it-' 


LIBRARY 

OF   THE 

UNIVERSITY  OF  CALIFORNIA. 

Received        |ai\j      f^    ^^93     .  /^^ 
^Accessions  No.HQon^,  class  No. 


^F#*; 


,V-K 


•-:e^i^¥-^*-:. 


» '/r.-^. 


^^= 


*  IT-/:;- 


«  » 


^^***v^     


ft 


.«•» 


♦  ? 


■.-  4  t      ;v  •  ,'•.:/  «  i      X  ■       ;.•   a  * 


Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/formlessonstopreOOspeerich 


FORM  LESSONS. 


TO  PREPARE  FOR  AXI)  TO  ACCOMPANY 


The  Study  of  Number. 


F 


BY 

AY,  ^y.  SPEER 

TEACHEIi  OK  iMATHEMATICS,  COOK  COUXTV  NOKMALi  SCHOOL,. 

THIRD  EDITION. 


C.  C.  N.  S.    SERIES. 


\.-<<  \ 


s 


r-1 


COPYRIGHT, 
18«8. 
BY  W.  W.  SPEER. 


DONOHUE  &  HENNEBERRV 

PRINTEilS  AND    BINDERS 

CHICAGO. 


PREFACE. 


In  the  first  grade,  this  "book  is  designed  to  aid  teachers  in  sys- 
tematizing their  oral  lessons.  In  the  other  grades,  the  book  should 
be  in  the  hands  of  each  pupil.  The  constant  test  of  the  pupil's  obser- 
vation must  be  the  accuracy  of  his  expression.  The  questions  asked 
arc  for  the  pupil  to  answer,  and  not  the  teacher.  When  his  descrip- 
tions are  not  correct,  he,  through  renewed  observation,  must  discover 
and  correct  his  error.  It  is  not  expected  that  all  that  can  be  dis- 
covered in  tlie  various  forms  will  be  seen  by  first  or  second-year 
pupils,  but  many  tests  have  shown  that  the  work  is  well  adapted  for 
these  grades,  while  it  furnishes  sufficient  material  to  profitably  engage 
the  attention  of  pupils  of  any  grade.  As  an  aid  to  mathematical 
investigation  and  for  securing  clear  and  concise  expression,  it  will,  I 
think,  be  found  helpful  even  in  the  high  school  before  beginning  the 
study  of  scientific  geometry,  if  pupils  have  not  had  previous  training 
of  this  character. 

Where  the  programme  is  so  full  that  there  is  not  time  to  give 
special  attention  to  teaching  form,  the  lessons  can  be  used  as  a  means 
of  teaching  language.  As  the  forms  to  be  compared  are  definite,  and 
demand  accuracy  of  expression  in  description,  and  as  all  inaccuracy 
of  statement  can  be  readily  detected,  no  study  furnishes  a  better  basis 
for  language  lessons.  In  the  written  exercises,  penmanship,  spell- 
ing, and  punctuation  can  be  taught. 

The  attention  of  the  reader  is  respectfully  invited  to  a  consid- 
eration of  the  remarks  on  page  73,  on  the  value  of  form  lessons  as  a 
preparation  for  number  study,  and  to  the  opinions  of  some  eminent 
educators  and  thinkers  on  this  subject. 

Englewood,  III.  W.  W.  SPEER. 


OBSERYATION   LESSON'S. 


"  Lessons  especially  designed  to  cultivate  the  power  and  habit 
of  oo^ervation  appear  to  be  less  widely  used  than  is  desirable.  It  is 
probable  that  the  erroneous  methods  of  teaching,  too  often  employed 
in  such  lessons,  have  led  to  meager  results  and  consequent  distrust 
of  this  branch  of  primary  school  work.  The  skill  required  to  teach 
such  lessons  properly  is  apparently  less  common  than  skill  in  teach- 
ing other  branches.  The  mistake  often  made  is  that  of  supposing  a 
pupil  is  learning  how  to  observe  when  he  is  merely  listening  to  what 
his  teacher  tells  him  to  remember  about  an  object  he  may  be  looking 
at.  So-called  object  lessons  taught  by  such  false  methods  have  no 
tendency  to  cultivate  the  power  and  habit  of  observation,  but  rather 
to  confuse  and  stultify  the  child's  mind.  On  the  other  hand,  obser- 
vation lessons,  in  which  the  children  really  do  the  observing,  not  only 
develop  the  observing  powers,  but  also  furnish  the  children's,  minds 
with  a  stock  of  clear  ideas  which  constitute  the  best  possible  material 
for  language  work." 

— Report  of  Massachusetts  Teachers'  Association,  1887. 


w 


LESSONS  IN  FORM. 


TO  PREPARE  FOR  AND  TO  ACCOMPANY 


THE   STUDY  OF  NUMBER. 


PART    FIRST. 

The  first  exercises  are  to  cultivate  the  habit  of 
observation,  to  teach  direction  and  position,  and  to 
train  pupils  to  associate  the  terms  to  be  used  in  the 
comparisons  which  follow  with  the  corresponding 
ideas. 

SURFACES,  LINES  AND  POINTS. 


Give  each  pupil  a  cubic  inch,  or  have  a  large  cube 
placed  so  that  the  class  can  observe  it. 

Directions  and  Questions  for  Pupils : 

Place  your  finger  on  the  surface  of  tlie  cube. 
Place  voui'  hand  on  the  surface  of  vour  desk. 


6  LESSONS   IN    FORM. 

Of  what  object  do  you  see  the  surface  ? 

Kecall  objects  that  you  have  seen  at  home  that 
have  surfaces. 

Find  other  surfaces  in  the  room. 

Of  what  object  at  home  are  you  thinking  that  has 
a  surface  ? 

Phice  your  linger  on  one  of  the  edges,  or  lines,  of 
the  cube.     On  one  of  the  edges,  or  lines,  of  your  desk. 

Find  other  lines,  or  edges,  in  the  room. 

Of  what  have  3^ou  found  an  edge,  or  a  line? 

Name  five  objects  that  you  have  seen  outside  of 
the  school-room  that  have  edges  or  lines. 

I^ame  five  objects  that  3^ou  have  seen  outside  tiie 
school-room  whose  surface  is  not  bounded  by  lines. 
Example :  The  surface  of  a  croquet  ball  is  not  bounded 
by  lines. 

Drawing. — Look  at  the  cu])e  and  draw  four  lines 
each  as  long  as  an  edge  of  the  cube. 

Use  an  edge  of  the  cube  and  measure  the  lines 
you  have  drawn. 

Draw  again  and  measure. 

Observe  a  face  of  the  cube  and  draw  it.  Continue 
drawing  and  measuring  until  you  can  represent  a  face 
quite  accurately. 

Place  a  finger  on  one  of  the  corners,  or  points,  of 
the  cube.     On  a  corner,  or  point,  of  your  desk. 

Find  other  points  in  the  room. 

Of  what  have  you  found  a  point  ? 

Place  a  finger  on  one  of  the  lines  of  the  cube. 

What  is  at  the  end  of  this  line  ? 

Can  you  place  a  finger  on  a  point  which  is  at  the 
end  of  one  of  the  lines  of  the  blackboard  ? 


OPPOSITE,    ADJACENT   AND    PARALLEL    LINES.  7 

OPPOSITE,  ADJACENT  AXD  PAPtALLEL  LINES.* 

The  upper  and  lower  lines,  or  edges,  of  the  black- 
board are  opposite  lines  of  the  blackboard.  Can  you 
find  other  edges,  or  lines,  of  the  blackboard  that  are 
opposite  ? 

Point  to  the  other  lines  of  the  blackboard  that  are 
opposite. 

Place  your  finger  on  two  lines  of  the  cube  that  are 
opposite. 

Find  opposite  lines  of  your  book. 

Find  other  opposite  lines  in  the  room. 

Of  what  have  you  found  opposite  lines? 

Drawing. —  Look  at  two  of  the  opposite  lines  of 
the  cube  and  draw  two  lines  Jis  long  and  as  far  apart. 

Measure  and  see  if  they  are  the  same  length  and 
as  far  apart  as  the  opposite  lines  in  tlie  face  of  the 

cube.     Draw  again  and  measure. 

In  Avhat  direction  do  the  opposite  lines  of  the  cube 
extend  ? 

Find  other  lines  in  the  room  that  extend  in  the 
same  direction. 

Of  what  have  you  found  lines  extending  in  the 
same  direction? 

Do  the  lines  you  drew  extend  in  the  same  direc- 
tion? 

Find  two  lines  of  the  blackboard  that  meet. 

Touch  two  edges,  or  lines,  of  the  cube  tliat  meet. 

*If  words  are  taught  in  connection  with  the  ideas,  primary  inipils 
will  have  no  difTiciilty,  because  of  their  lensrth,  in  learning'-  to  use  iniraliel, 
parallelo^rram,  perimeter,  rectangle,  rectangular,  blackboard,  grandlafher, 
or  parallelopipcd.  It  is  not  wise  to  teach  one  set  of  terms  in  theiirimary 
and  another  in  the  higher  grades.  Teach  the  right  terms  from  the  begin- 
ning, so  that  i)upils  may  learn  to  think  in  the  proper  language. 


8  LESSOITS   IN   FORM. 

Lines  that  meet  are  adjacent  lines.  Find  other 
adjacent  lines  in  the  cube. 

Find  other  adjacent  lines  in  the  room. 

Of  what  have  you  found  adjacent  lines,  or  edges  ? 

Deaaving. —  Draw  a  pair  of  adjacent  lines,  making 
each  line  an  inch  long.  Draw  another  pair,  making 
each  hne  two  inches  long.  Draw  another  pair,  making 
each  hne  three  inches  long. 

Measure  each  pair  and  draw  again. 

Find  opposite  lines  and  adjacent  lines  in  the  room 
and  tell  whether  thej  are  opposite  or  adjacent.     Ex- 

ample :     This  line  and  that  line  of  the  door  are  oppo- 
site. 

Lines  that  extend  in  the  same  direction  are  par- 
allel. Find  parallel  lines  of  the  cube.  Of  a  book.  Of 
a  chalk  box. 

Drawing. —  Draw  three  pairs  of  parallel  lines  each 
an  inch  long.     Measure  and  draw  again. 

Do  lines  need  to  be  of  the  same  length  that  they 
may  extend  in  the  same  direction  ? 

Do  they  need  to  be  of  the  same  length  that  they 
may  be  parallel  ? 

"When  are  lines  parallel  ? 

Can  you  find  opposite  lines  in  the  room  that  are 
not  parallel  ? 

Suggestion:  If  pupils  can  not  find  opposite  lines  in  the  room 
that  are  not  parallel,  draw  several  quadrilaterals  on  the  blackboard 
of  which  one  or  both  pairs  of  opposite  lines  do  not  extend  in  the 
same  direction.  Have  pupils  point  to  lines  of  the  figures  that  are 
opposite  but  not  parallel. 


ANGLES   AND   PERPENDICULAR   LINES.  9 

ANGLES  AND  PERPENDICULAR  LINES. 


TlILSE  ARE  PeKFENDTCCLAII  LiNES. 

Find  lines  in  tlie  r<,.oni  tliiit  aro  perpendicular  to 
each  other. 

How  many  pairs  of  perpendicular  lines  are  there 
in  the  blackboard  ? 

Ans.  The  edges  of  tiie  blackboard  form  four  pairs 
of  perpendicular  lines. 

How  many  pairs  of  perpendicular  lines  are  there 
in  one  face  of  the  cube  ? 

How  many  pairs  of  opposite  lines  are  there  in  one 
face  of  the  cube  ? 

How  many  pairs  of  opposite  lines  are  there  in  one 
of  the  Avails  of  the  room  I 

Remaek  :  "When  the  attention  of  pupils  is  first  called  to  the 
fact  that  lines  may  extend  in  different  directions,  or  form  angles, 
you  will  do  them  a  service  by  using  for  the  first  illustration  lines 
that  do  not  intersect.  A  majority  of  pupils,  who  think  they  know 
what  an  angle  is,  are  very  sure  that  unless  the  lines  do  intersect 
there  is  no  angle.  This  prejudice  would  not  exist  had  their  first  im- 
pression of  an  angle  hecn  gained  when  considering  the  difference  in 
direction  of  lines  that  did  not  intersect. 

Point  to  lines  extending  in  the  same  direction. 
Point  to  lines  that  extend  in  different  directions. 
When  two  lines  extend  in  different  directions,  the 
difference  in  direction  is  an  angle. 


10  LESSON'S   IN   FORM. 

Find  five  angles  in  the  room.  When  you  point  to 
an  angle,  say  that  the  difference  in  direction  of  this  line 
and  that  line  (indicating  the  direction  each  line  extends, 
by  moving  pointer  or  hand)  is  an  angle. 

Hold  two  sticks  or  slats  so  that  they  are  perpen- 
dicular to  each  otlier. 

The  difference  in  direction  of  two  perpendicular 
lines  is  a  right  angle. 

What  is  a  rio;ht  ang-le  ? 

Find  perpendicular  lines,  and  say  that  the  differ- 
ence in  direction  of  these  perpendicular  lines  is  a  right 
angle. 

Hold  two  splints  so  that  the  difference  in  direction 
which  they  indicate  is  not  so  great  as  a  right  angle. 

When  the  difference  in  direction  of  two  lines  is  not 
so  great  as  that  of  a  right  angle,  the  difference  in 
direction  is  called  an  acute  angle. 

Find  m  the  room  differences  in  direction  less  than 
a  right  angle.  What  is  a  difference  in  direction  less 
than  a  rio^ht  anoxic  called  ? 

Hold  the  slats  or  splints  so  that  the  difference  in 
direction  is  greater  than  a  right  angle. 

When  the  difference  in  direction  of  two  lines  is 
greater  than  the  difference  in  direction  of  two  perpen- 
dicular lines  it  is  called  an  obtuse  angle. 

What  is  an  obtuse  angle? 

Ans.  An  obtuse  angle  is  a  difference  in  direction 
greater  than  a  right  angle. 

Find  obtuse  angles  in  the  room  and  tell  Avhy  they 
are  obtuse.  Example:  The  difference  indirection  of 
this  line  and  that  line  is  an  obtuse  angle  because  it  is 
greater  than  a  right  angle. 


ANGLES   AND   PERPENDICULAR   LINES.  11 

How  many  differences  in  direction  can  you  repre- 
sent with  two  lines?     With  three  hnes?     With  four 

hnes? 

Represent  three  right  angles  by  drawing  lines. 

Can  \^ou  draw  two  right  angles  so  that  the  differ- 
ence in  direction  of  one  is  greater  than  the  difference 
in  direction  of  the  other  ? 

If  you  make  long  lines  in  representing  a  right  an- 
gle, is  the  difference  in  direction  greater  than  when 
represented  by  short  lines  ? 

Why  not^? 

Can  you  make  two  right  angles  with  two  lines? 

Can  you  make  three  right  angles  with  two  lines? 

Can  you  make  four  with  two  lines  ? 

How  many  right  angles  can  you  make  with  three 
lines? 

Draw  tliree  acute  angles. 

Can  you  draw  two  acute  angles  so  that  the  differ- 
ence in  direction  in  one  shall  be  greater  than  the  differ- 
ence in  direction  of  the  other  ? 

Draw  three  acute  angles  of  different  sizes. 

Draw  three  obtuse  angles. 

Can  you  draw  obtuse  angles  of  different  sizes,  or 
magnitudes  ? 

If  you  increase  the  length  of  the  lines  of  an  obtuse 
angle  does  it  increase  the  difference  in  direction? 
Does  it  increase  the  size  oP  the  angle  ?     Why  not? 

There  are  what  three  kinds  of  angles  ? 

What  is  a  rio"ht  ano-le?  An  acute  anHe?  An 
obtuse  angle  ?  "^" 

*  IlKMAUKS :  Two  straight  lines  may  have  cither  of  two  relative  posi- 
tions . 

1st.    Tlicy  may  extend  in  the  same  flireetion. 
2(1.     They  mai'  extcinl  in  <lifrorent  directions. 


>. 


V   r* 


12  LESSONS   IN   FORM. 

DIEECTION  AND  POSITION. 


Place  your  linger  on.  the  front  face  of  the  cube, 
that  is,  the  face  next  to  you.     On  the  back  face.     On 

In  the  first  case,  instead  of  saying  that  the  lines  extend  in  the  same 
direction,  it  is  customary  to  say  the  lines  are  parallel.  One  form  of 
expressing-  this  relation  means  no  more  nor  less  than  the  other.  Parts  of 
the  same  line  thought  of  as  distinct  lines  are  parallel,  for  they  can  be  thought 
of  as  extending  in  the  same  direction. 

In  defining  parallel  lines,  do  not  add  either  of  the  following  state- 
ments to  the  definition,  namely  :  If  produced,  they  would  never  meet ;  or, 
they  are  everywhere  equally  distant.  It  is  not  true  that,  when  the  parallels 
are  parts  of  the  same  straight  line,  they  would  never  meet  if  produced,  and, 
it  is  an  inference,  from  the  definition  proper,  that  they  would  never  meet  if 
they  are  not  paints  of  the  same  line.  An  inference  should  not  be  a  part  of  a 
definition.  That  parallels  are  everywhere  equally  distant,  if  they  are  rot 
parts  of  the  same  straight  line,  is  a  proposition  to  be  demonstrated  in  geom- 
etry, and  it  ought  not  to  be  made  a  part  of  a  definition. 

The  second  relation  of  lines,  that  of  extending  in  different  directions, 
is  called  an  angle.  An  angle  is  simply  a  difference  in  direction.  It  may  be 
a  difference  in  direction  of  two  lines,  two  or  more  surfaces,  of  two  persons 
walking,  etc.  It  is  not  necessary  that  the  lines  intersect  in  order  that  they 
indicate  a  difference  in  direction.  Nothing  is  added  to  the  difference  in 
direction  by  the  intersection.  If  the  lines  forming  the  angle  do  not  inter- 
sect, and  are  in  the  same  plane,  it  is  an  inference  that  they  will  intersect  if 
produced  in  the  direction  of  their  convergence,  and  another  inference  that, 
if  produced  in  the  direction  of  their  divergence,  they  will  not  intersect. 

In  orderto  measui-e  an  angle,  or  to  pi-ove  it  equal  to  another  angle  by 
superposition,  the  lines  indicating  the  difference  in  direction  must  inter- 
sect, and,  therefore,  lie  in  the  same  plane. 

The  importan  e  of  fixing  right  ideas  of  the  two  relations  of  lines  will 
be  recognized  when  it  isimderstood  that  all  of  the  reasoning  in  both  plane 
and  solid  geometry  is  based  on  a  comparison  of  the  length  of  lines,  the 
sameness  of  direction  or  difference  in  direction  of  lines.  Hazy  ^'mpressions 
of  these  elementary  ideas  will  make  the  reasoning  based  on  such  impressions 
lack  in  clearness. 


SURFACES,  LIKES  AND    POINTS   OF   A   CUBE.  13 

the  left-hand  face.  On  the  upper  face,  or  base.  On 
the  right-hand  face.  On  the  upper  right-hand  corner 
in  front.  On  the  upper  left-hand  corner  at  the  back. 
On  the  lower  left  hand  corner  in  front.  On  the  upper 
edge  of  the  front  face.  On  the  upper  edge  of  the  left- 
hand  face. 

Suggestions:  Have  one  pupil  give  a  direction  for  pointing  or 
placing  a  finger^  and  other  pupils  follow  directions.  Have  other 
pupils  give  similar  directions. 

Have  pupils  point  to  different  corners  of  (.bjects  ana  tell  "where 
each  is.  Example:  That  is  the  upper  right  hand  corner  in  the  front 
part  of  the  room. 

Place  a  finger  on  different  points  and  faces  of  the  cube  or  other 
objects,  and  have  pupils  tell  where  it  is. 

Have  pupils  point  to  different  corners  of  the  room  and  tell  to 
what  corner  they  are  pointing.  To  different  windows  in  the  room 
and  tell  where  each  is.  ExamjJle:  That  is  the  east  window  in  the 
north  wall.  To  different  pupils  and  tell  where  each  is.  Examjle: 
Mary  Warner  sits  on  the  third  seat  in  the  second  row  from  the  right. 

Review  exercises  until  pupils  can  follow  directions  in  j^lacing 
finger  on  face,  line,  or  point  designated,  and  can  give  directions 
clearly  and  without  hesitation. 


THE  SUEFACES,  LIXES  AND  POINTS  OF  A  CUBE. 


Count  the  surfaces  of  a  cubic  inch. 
How  many  surfaces  has  a  cubic  inch? 


14  LESSONS    IN    FORM. 

How  many  surfaces  has  a  chalk  box  ? 

Find  other  objects  in  the  room  that  have  six 
surfaces. 

What  object  have  you  found  that  has  six  surfaces  ? 

What  objects  have^^ou  seen  at  home  or  in  going  to 
and  from  the  school  that  have  six  surfaces? 

How  many  surfaces  has  the  school-room  ? 

(Count  the  Avails,  ceiling  and  floor.) 

Place  a  finger  on  the  upper  face,  or  base,  of  the 
cubic  inch. 

Count  the  edges  of  the  upper  base. 

How  many  edges  has  the  upper  base  ? 

How  many  edges  has  the  lower  base  ? 

Point  toward  the  zenith,  or  the  point  in  the 
heavens  directly  over  3^ our  head. 

Point  toward  the  center  of  the  earth.' 

An  upright  line,  or  a  line  which  extends  from  the 
zenith  toward  the  center  of  the  earth,  is  a  vertical  line. 

From  what  and  toward  what  does  a  v^ertical  line 
extend  ? 

Find  vertical  lines  in  the  school-room.  Example : 
One  of  the  right-hand  edges  of  the  door  is  an  up-and- 
down  line,  or  a  vertical  line. 

Kecall  objects  that  are  in  a  vertical  position.  Ex- 
amjple :  Some  telegraph  poles  are  in  a  vertical  posi- 
tion. 

Can  you  find  edges  of  the  cube  that  are  vertical  ? 

How  many  lines  of  the  cube  are  vertical  ? 

Point  toward  the  horizon. 

Find  lines  in  the  room  that  extend  toward  the 


SURFACES,    LINES    AXD    POIXTS    OF   A    CUBE.  15 

horizon.  Exami)le :  The  upper  edge  of  the  blackboard 
extends  toward  the  horizon. 

Lines  that  extend  towara  the  horizon  are  horizon- 
tal lines.  Find  horizontal  lines  in  the  room.  AVhat 
lines  of  a  window  are  horizontal? 

In  what  direction  do  horizontal  lines  extend? 

If  you  place  a  cube  so  that  four  of  its  edges  ai"e 
vertical,  how  many  horizontal  edges  will  it  have? 

Can3^ou  hold  a  cube  so  that  none  of  its  edges  will 
be  parallel?  Why  not?  Can  you  hold  a  cube  so  that 
none  of  its  edges  will  be  perpendicular  to  each  other? 

Does  it  change  the  relation  of  the  lines  of  a  cube 
to  each  other  to  hold  the  cube  indifferent  positions? 

Can  3^ou  hold  a  cube  so  that  its  edges  will  be 
neither  vertical  nor  horizontal? 

How  many  lines  has  the  cubic  inch  ? 

How  many  lines  has  a  cubic  foot? 

Kame  other  objects  that  have  twelve  lines,  or 
edges. 

How  many  lines  bound  the  ceiling  of  the  room  ? 

How  many  lines  bound  the  floor  of  the  room? 

How  many  vertical  edges  have  the  walls  of  the 
room  ? 

How  many  corners,  or  points,  has  the  cubic  inch  ? 

How  many  points  has  the  upper  base? 

How  numy  p(jints  has  the  lower  base? 

How  many  points  has  a  cubic  foot  ? 

How  many  points  has  a  brick  ? 

How  many  lines  meet  in  one  of  the  points  of  the 
cubic  inch? 

In  how  many  directions  do  the  three  edges  extend 
from  one  point  i 


16  LESSORS  i:n'  porm. 

How  many  pairs  of  parallel  lines  are  there  in  one 
face  of  the  cube  ? 

How  many  surfaces  has  the  cube? 

How  many  two  pairs  of  parallel  Lnes  has  the 
cube? 

How  many  square  inches  are  there  in  the  surface 
of  the  cubic  inch  ? 

Can  you  tell  eight  things  that  are  true  of  the  cubic 
inch  ? 

What  is  the  shape  of  one  of  the  surfaces  of  the 
cubic  inch  ? 

Find  otlier  squares  in  the  room. 

What  object  have  you  found  that  has  a  square  sur- 
face? 

Can  you  recall  any  squares  that  3^ou  have  seen  at 
home  ?  Examples :  The  tops  of  some  collar  boxes  are 
square.     The  bottoms  of  some  salt  cellars  are  square. 

Can  you  recall  any  squares  that  3^ou  have  seen  in 
going  to  and  from  school  ? 

Review,  beginning  with  surfaces,  lines  and  points 
of  a  cube. 

Dkawing. — Draw  a  two-inch  square.  Use  a  ruler 
and  measure.  Erase.  Draw  and  measure  again.  Erase. 
Use  the  ruler  and  draw  a  two-inch  square.^ 

*  Remark  :  Manj^  of  the  pupils,  in  their  efforts  to  make  two-inch 
squares  or  to  represent  anything-,  will  not  do  well  at  first  if  the  drawing-  and 
not  the  effort  is  considered.  Tlie  effort  made  to  represent  the  thing 
observed  or  recalled  is  worth  a  thousand  times  more  than  the  drawing, 
■which  is  the  result  of  the  effort.  The  question  is  not,  Are  they  making  ac- 
curate and  beautiful  drawings?  but,  ai'e  they  forming  habits  of  observa- 
tion ? 

Drawing  furnishes  a  means  of  expressing  ideas,  and  man  first  resorted 
to  it  for  that  purpose;  but  when  it  is  perverted  and  fails  to  accomplish  this 
purpose,  it  does  not  produce  the  best  results.    Any  method  that  teaches 


QUESTIONS   OlST   THE   TWO-IXCH   SQUARE.  17 

THE  SQUAEE. 


c  d 

Tell  as  many  things  as  you  can  about  iliu  two-inch 
square. 

QUESTIONS  ON  THE  TWO-INCH  SQUARE. 

1.  How  many  lines  bound  the  square? 

2.  How  many  points  has  the  square? 

3.  How  many  pairs  of  parallel  lines  has  the 
square?     How  do  parallel  lines  extend  ? 

words  before  ideas  is  radically  wrong-,  ami  any  method  that  teaches  dra  wing- 
without  usinjr  it  as  a  means  of  expressinj^or  representing-  ideas,  is  radically 
wrong-,  because  it  leaves  out  that  whch  stimulates  and  dovch  ps  the  powers 
of  the  mind.  Iicpruducin^- aline  Avithouta,considcring-  its  leu'rth  or  direc- 
tion does  very  little  to  increase  one's  power.  That  traininj?  which  leads 
puynls  to  be  imitators  only  does  little  to  develop  thought  and  action. 
Drawing- ought  to  teach  seeing,  doing,  and  knowing.  Drawing  ought  to  cul- 
tivate the  hand  and  the  eye,  and  increase  the  knowledge  of  the  object 
represented. 

'•  As  the  first  step  in  drawing  is  to  learn  to  see  correctly,  it  is"evident 
that  all  the  exercises,  both  in  gifts  and  occupations,  prepare  for  the  use  of 
the  pencil  and  chalk.  As  the  mediation  of  word  and  object  drawing  is  of 
vast  importance  in  its  reaction  on  tl-.e  iiiind,  and  as  th(>  soul  of  all  technical 
processes,  it  is  the  indispensable  basis  of  industrial  education." 

Susan  E.  Blow, 


18  LESSONS   12^   FORM. 

4.  How  many   pairs   of   adjacent   lines   has  the 
square  ?     (Each  line  is  counted  twice.) 

5.  How  many  pairs   of  perpendicular   lines   are 
there  in  the  square  ? 

G.     The  difference  in  direction  of  two  perpendicu- 
lar lines  is  what  kind  of  an  angle? 

7.  How  many  right  angles  are  there  in  the  square  ? 

8.  How  many  inches  in  the  boundary,  or  perime- 
ter, of  the  two-inch  square  ? 

"  That  which  the  pupil  knows  thoroughly  contains 
an  explanation  of  what  he  does  not  know." 

DIRECTION  AND  POSITION. 
To  Teacher:    Give  eacli  pupil  a  4  inch  square,  and  have  him 
place  the  letter  a  rear  the  upper  left-hand  point  at  the  back,  b  near 
the  right-hand  point  at  the  back,  c  near  the  left-hand  point  in  front, 
and  d  near  the  right-hand  point  in  front. 

QUESTIONS. 

1.  In  what  part  of  the  square  is  the  point  a  ?    Ans. 
The  point  a  is  in  the  left-hand  point  at  the  back. 

2.  Where   is  the  point   <??     The   point   J?     The 
point  cPi 

3.  Where  is  line  a  h  of   the  square  ?     Ans.     The 
line  ah  is  the  line  at  the  back  of  the  square. 

4.  Where  is  the  line  c  d\     The  line  a  c  ?     The  line 
ld\ 

5.  The  points  c  and  h  are  opposite  points  of  the 
square.     Wha.t  other  points  of  the  square  ai"e  opposite  ? 

6.  The  line  a  h  extends  in  the  same   direction  as 
what  line  ? 

Y.     What  are  lines  called  that  extend  in  the  same 
direction  ? 


QUESTIONS    ON   THE   TWO-INCH   SQUARE.  19 

Suggestion:  Draw  a  square  foot  on  the  blackboard.  Write 
the  letter  a.  near  the  upper  left-hand  point,  Z>  near  the  upper  right- 
hand  point,  ('.  near  the  lower  left-hand  point,  and  d  near  the  lower 
right-hand  point. 

1.  Which  is  the  point  a  \  Ans.  It  is  the  upper 
left-hand  point  of  the  sq.  ft. 

2.  Which  is  the  point  dl  The  point  M  Tlie 
point  G% 

3.  Which  is  the  line  (3^  h  of  the  sq.  ft.  % 

4.  Which  is  the  line  a  c  oi  the  sq.  ft.  ? 

5.  Close  your  eyes.  Near  what  point  is  each  let- 
ter in  the  sq.  ft.  ?  The  letter  d  is  near  the  lower  right- 
hand  point  of  the  sq.  ft. 

Suggestion:  Have  pupils  tell  as  many  different  things  as  they 
can  about  the  positions  of  lines  and  points  of  the  blackboard,  door, 
etc.,  without  questioning  them. 

6.  Call  the  edge  of  your  desk  next  to  you  the 
front  edge. 

7.  Place  your  hand  on  the  front  edge  of  your 
desk.  On  th  ^Ag'^  at  the  back.  On  the  middle  of  the 
right  edge. 

8.  Toucn  tlie  left  corner  in  front.  The  right  cor- 
ner at  the  back.  Place  your  hand  on  the  middle  of  the 
edge  at  the  back. 

Suggestion  to  Teacher:  Have  pupils  give  directions  for 
other  pupils  to  follow.  Giving  directions  will  force  pupils  to  express 
themselves  with  precision.  The  necessity  of  saying  exactly  what 
they  mean  will  make  the  exercise  a  valuable  language  lesson. 

REVIEW    KROM   THE   BEGINNING   OF   THE  BOOK. 


20 


LESSONS   IN   FORM. 


ExEEOiSES  IN  Compaeiso:n.* 


COMPARISON  OF  THE  SQUARE  RECTANGLE  WITH  THE 
OBLONG  RECTANGLE. 

Remaek:     The  lieavy  lines,  iu  the  different  cuts,  indicate  the 
forms  to  be  compared  made  by  folding  the  square. 

Get  kindergarten  4-inch  squares,  or  cut  4-inch  squares  out  of 
paper.     Give  each  pupil  two  of  these  squares. 


*"  It  is  iDy  comparisons  that  we  ascertain  the  difference  which  exists 
between  things,  and  it  is  by  comparisons,  also,  that  we  ascertain  the  general 
features  of  thing-s,  and  it  is  hy  comparisons  that  we  reach  general  proposi- 
tions. In  fact,  comparisons  are  at  the  bottom  of  all  philosophy.  Without 
comparisons  we  never  could  go  beyond  the  knowledge  of  isolated,  discon- 
nected facts.  Now,  do  you  not  see  what  importance  there  must  be  in  such 
training,— how  it  Avill  awaken  the  faculties,  how  it  will  develop  them,  how 
it  will  be  suggestive  of  further  inquiries  and  further  comparisons;  and  as 
soon  as  one  has  begun  that  sort  of  study  there  is  no  longer  any  dullness  in 
it.  Once  imbued  with  the  delight  of  studying  the  objects  of  nature,  the 
student  only  feels  that  his  time  is  too  limited  in  proportion  to  his  desire  for 
more  knowledge.  And  I  say  that  we  can  in  this  way  become  better 
aquainted  with  ourselves.    .    .    . 

"  The  difficult  art  of  thinking  can  be  acquired  by  this  method  in  a  more 
rapid  way  than  any  other.  When  we  study  logic  or  mental  philosophy  in 
text- books,  which  we  commit  to  memory,  it  is  not  the  mind  which  we  culti- 
vate ;  it  is  the  memory  alone.  The  mind  may  come  in,  but  if  it  does  in  that 
method  it  is  only  in  an  accessory  way.  But  if  we  learn  to  think,  by  unfold- 
ing thoughts  ourselves  from  the  examination  of  objects  brought  before  us, 
then  we  acquire  them  for  ourselves,  and  we  acquire  the  ability  of  applying 
our  thoughts  in  life."  Agassiz. 


COMPARISON    OF    RECTANGLES.  21 

Direction  for  folding  the  square  into  the  oblong 
rectangle: 

Place  one  of  the  squares  so  that  its  front  edge  will 

extend  in  the  same  direction  as  the  front  edge  of  the 

desk.     Fold  the  paper  so  that  the  front  edge  will  lie 

along  or  coincide  with  the  edge  at  the  back.     Crease 

the  paper.     Place  the  oblong  rectangle  formed  near  the 

square  for  comparison. 

Remark:  When  the  pupils  are  folding  papers  they  should  be 
ri'quired  to  keep  themiu  the  same  relative  position.  If  they  do  not 
they  can  not  follow  directions.  They  should  not  lift  the  paj)ers  oU 
the  desks  when  folding. 

1.  Find  the  likenesses. 

2.  Find  the  differences. 

3.  Find  square  rectangles  in  room. 

4.  Recall  square  rectangles  that  you  have  seen  at 
home  or  in  going  to  and  from  school. 

5.  Find  oblong  rectangles  in  the  room.  Example: 
One  of  the  windows  is  the  shape  of  an  oblong  rect- 
angle. 

6.  Recall  oblong  rectangles  that  you  have  seen. 
Example:  One  of  the  surfaces  oL'  a  brick  is  an  oblong 
rectangle. 

7.  In  going  to  and  from  school  or  at  home,  you 
may  find  five  square  rectangles,  and  five  oblong  rect- 
anoles,  and  in  to-morrow's  recitation  vou  mav  tell  me 
the  names  of  the  objects  that  have  square  surfaces  and 
those  that  have  oblong  rectangular  surfaces. 

The  following  questions  are  to  be  answered  by 
pu])ils  aft^r  they  have  found  all  the  likenesses  and  dif- 
ferences they  can  in  comparing  tlie  forms  above: 

Remark:  Pupils  should  bo  cncouraired  to  give  the  names 
of  the  colors  of  the  dilfereut  papers  used  in  the  comparisons  and  to 


22  LESSOKS   IK   FORM. 

find  like  colors  in  the  room,  and  to  recall  objects  that  are  similar  in 
color. 

QUESTIONS— LIKENESSES. 

1.  Each  of  the  forms  is  what  ? 

Ans.  Each  of  the  forms  is  a  surface,  or  they  are 
each  surfaces. 

2.  Each  surface  has  how  many  lines? 

3.  Each  has  how  many  points? 

4.  Each  has  how  many  pairs  of  parallel  lines? 

5.  Each  has  how  many  pairs  of  adjacent  lines  ? 

6.  Each  has  how  many  pairs  of  perpendicular 
lines  ? 

7.  Each  has  how  many  right  angles  ? 

8.  Each  bas  how  many  pairs  of  opposite  points? 

9.  One  of  the  long  lines  of  the  oblong  rectangle 
is  equal  to  what  in  the  square? 

10.  The  sum  of  the  tw^o  short  lines  of  the  oblong- 
rectangle  is  equal  to  what  ? 

11.  The  opposite  lines  in  each  are  what  ? 

12.  Give  the  likenesses  without  being  questioned. 

Remarks  :  In  elementary  work,  it  is  not  well  to-spend  time  ar- 
guing with  pupils,  or  trying  to  force  your  views  upon  them,  even 
if  you  are  in  the  right.  You  can  not  correct  false  notions  or  nar- 
row views  unless  there  arc  ideas  enough  in  the  pupil's  mind  to  en- 
able him  to  comprehend  what  you  s^y.  If  you  develop  the  dis- 
criminating power  of  the  pupil  by  training  him  to  observe,  he  will, 
in  time,  see  for  himself  what  you  wish  him  to  see.  In  observing 
the  square,  one  pupil  may  see  on]y  two  pair  of  perpendicular  lines, 
while  another  sees  the  four.  The  former  thinks  that,  as  there  are 
only  four  lines  in  the  square,  there  can  not  be  more  than  two  pair 
of  perpendicular  lines.  Trying  1o  convince  the  first  pupil  that  he  is 
wrong  may  be  worse  than  a  waste  of  time.  To-day,  he  sees  two  pair; 
leave  his  mind  free,  and  to-morrow  he  may  discover  the  four. 


COMPARISOIT   OF   RECTANGLES.  23 

DIFFEEEIS^CES. 

1.  The  square  rectanole  is  bounded  l)y  how 
many  lines?  The  oblong  rectangle  is  bounded  by  two 
equal  long  lines,  and  by  what? 

2.  One  of  the  short  lines  of  the  oblong  rectangle 
is  equal  to  what  part  of  one  of  the  lines  of  the  square 
rectangle  ? 

3.  The  oblong  rectangle  is  equal  to  what  part  of 
the  square  rectangle? 

4.  The  sum  of  the  lines  of  tlie  oblong  rectangle 
is  equal  to  how  many  of  the  lines  of  the  square? 

5.  How  many  inches  are  there  in  tlie  boundary 
of  each  form  ? 

G.  Can  you  find  rectangles  in  the  room  that  are 
not  one-half  of  a  square  rectangle  ? 

7.  Are  the  short  sides  of  oblong  rectangles  always 
equal  to  one-half  their  long  sides  ? 

8.  What  is  the  direction  for  folding  the  square 
into  the  oblono:  rectano:le  ? 

Cutting. — Direction  for  pupils  : 

Cut  out  of  paper  an  inch,  a  two-inch,  a  three-inch, 
and  a  four-inch  square.  Bring  them  to  the  class  to- 
morrow. You  may  cut  and  measure  as  many  as  you 
wish  before  cutting  those  you  bring  to  the  class.  Pin 
the  squares  you  bring  to  the  class  together  and  write 
your  name  on  one  of  them.^ 

*  "  Thrc/iitili  their  own  productions  children  are  slowlj'  awakening  to 
facts  of  form  and  relations  of  number  and  led  to  clear  and  concise  use  of 
language."  Miss  Susan  Blow. 

"Almost  invariablj'  cliildren  show  a  strong  tendency  to  cut  out 
things  in  j  aper,  to  make,  to  build— a  propensity  which,  if  (Uily  encouraged 
and  directed,  will  not  only  i)icpare  the  wiiy  for  scientific  conceptions,  but 
will  develop  those  powers  of  manipulation  in  which  most  people  arc  so  de- 
ficient." Herbert  Spencer. 


24 


LESSON^s  i:n^  form. 


Drawing. — You  may  bring  to  the  class  to-morrow 
the  different  squares  mentioned  above,  drawn  on  paper. 
You  ma}^  draw  and  measure  for  a  while,  but  do  not 
measure  those  you  bring  to  the  class. 

"  What  we  try  to  represent  we  begiu  to  understand." 

Froebel. 

Remark  :  The  pupils  should  be  encouraged  to  cut  and  meas- 
ure and  to  draw  and  measure  many  squares.  Practice  in  this  trains 
the  hand  and  eye.  Very  little  skill  will  be  required  by  pupils  who 
cut  and  draw  only  one  square  of  each  dimension  given. 

These  exercises  may  seem  very  simple,  but  neither  a  boy  of  five 
nor  a  man  of  fifty  can  cut  or  draw  a  four-inch  square  who  has  not 
been  trained  to  observe. 

Upon  our  perceptions  of  form  and  of  extent  depends  the  cor- 
rectness of  our  ideas  of  objects  and  consequently  our  power  of  giv- 
ing true  descriptions  of  things,  of  their  location,  their  size,  the  rela- 
tions of  one  part  to  another,  etc.,  etc. 

Give  each  pupil  two  four-inch  squares.  Give  direc- 
tion for  folding  the  squares  into  an  oblong  rectangle. 
Place  the  rectangle  on  one  side  of  the  desk.  Take  the 
other  square  and  place  it  so  that  one  of  its  edges  Avill 
extend  in  the  same  direction  as  tiie  front  edge  of  the 
desk.  Fold  the  square  so  that  the  right-hand  point  in 
front  will  fall  upon  or  coincide  with  the  left-hand  point 
at  the  back.  Crease  the  paper.  Place  the  two  folded 
figures  near  each  other  for  comparison. 

COMPARISON  OF  THE  OBLONG  RECTANGLE  AVITH  THE 

TRIANGLE. 


RECTANGLES   AKD    TRIANGLES.  25 

^' The  mistake  often  made  is  that  of  supposing  a 
pupil  is  learning  how  to  ohserve,  when  he  is  merely  list- 
ening to  what  his  teacher  tells  him  to  rememher  about 
an  object  he  may  be  looking  at." 

1.  Find  likenesses. 

2.  Find  differences. 

3.  Find  triangles  in  room.'^ 

4.  Recall  forms  that  are  triangular  in  shape. 
Examjyh:  The  gables  of  some  houses  are  triangular. 
The  lateral  surfaces  of  pyramids  are  triangles. 

5.  In  going  to  and  from  scliool  or  at  home  \o\\ 
may  find  five  objects  that  have  triangular  surfaces,  and 
you  may  tell  me  the  names  of  the  objects  that  have 
these  surfaces. 

().  What  are  the  names  of  the  objects  that  you 
found  having  square  surfaces  ? 

7.  What  are  the  names  of  those  that  you  found 
having  oblong  rectangular  surfaces? 

QUESTIONS— LIKENESSES. 

Suggestion:  Have  pupils  write  the  answers  to  the  following 
questions  in  complete  sentences.  In  the  recitation  have  answers 
read,  and  let  pupils  criticise  one  another's  statements. 

1.     Each  form    is  what? 

♦Remark  :  Fin  'iii;^  formsof  t'nos  'mo/rener  dshape  as  thosetakon  jjs 
.types,  is  of  the  hij^liestiniixii-tanuc.  Uult-sstiiisi.sdc.ncpupilsaie  nut  learn- 
ing to  pass  from  the  partieuhiv  to  the  general.  They  are  not  taught  to  see 
many  things  through  the  one,  a  .d  the  impression  they  gain  is  that  the  par- 
ticular forms  observed  i  re  tlie  only  forms  of  this  kind.  Unless  that  which 
the  pupil  observes  ;  ids  him  in  interprctin.r  something  else,  it  is  ('f  no  val.ie 
to  him.  Certain  things  arc  taught  that  tlirouuh  them  (ther  things  may  be 
seen.  Pupilsshould  not  be  trained  tosee  forthesake  of  the  seeing, but  that 
they  may  liave  tlie  power  t)  see.  Il(;w  diU'crent  t!ie  world  up(.ears  to  a 
child  who  sees  form  in  everything  from  what  it  d.>fcs  to  one  who  sees  no 
definite  form  in  anything,  and  to  whom  all  is  in  a  stiite  of  confusion.  Teach- 
ing is  leading  pupils  to  discover  the  unity  of  things. 


\v     >% 


26  LESSONS    IN    FORM. 

Ans.     Each  form  is  a  surface. 

2.  Each  surface  is  bounded  hv  what? 

3.  The  two  short  lines  of  the  triangle  are  equal  to 
what  ? 

4.  The  sum  of  the  two  short  lines  of  the  oblong- 
rectangle  is  equal  to  what  ? 

5.  The  sum  of  the  two  acute  angles  of  the  triangle 
is  equal  to  what  ? 

6.  Each  form  has,  at  least,  one  pair  of  what  kind 
of  lines  ? 

7.  Each  form  has,  at  least,  one  angle  of  what  kind  ? 

8.  The  area  of  the  rectangle  is  equal  to  what  part 
of  the  four-inch  square  ? 

9.  The  area  of  the  triangle  is  equal  to  what  part 
of  the  four-inch  square  ? 

10.  The  area  of  the  triangle  is  equal  to  what? 

11.  Without  observing  the  forms  think  of  all  the 
likenesses  you  can. 

DIFFERENCES. 
Write  answers  in  complete  sentences. 

1.  How  many  lines  are  there  in  the  boundary  of 
each  surface  ? 

2.  Each  surface  has  how  many  pairs  of  adjacent 
lines  ? 

3.  Eacn  surface  has  how  many  angles  or  diifer- 
ences  in  direction? 

4.  Each  has  how  many  right  angles  ? 

5.  Each  has  how  man}^  acute  angles? 
Find  the  right  angle  of  the  triangle. 

Find  the  line  opposite  the  right  angle  of  the  tri- 
angle. The  line  opposite  the  right  angle  of  a  right  tri- 
angle is  the  hypotenuse  of  the  right  triangle. 


RECTANGLE    AXD   TKTAXGLE.  27 

6.  The  hypotenuse  of  a  right  angle  is  opposite 
what  ? 

7.  What  is  opposite  the  hypotenuse  of  a  right 
triangle  ? 

8.  The  hypotenuse  of  the  right  triangle  is  longer 
than  what  in  tlie  rectangle? 

9.  Is  there  any  line  opposite  another  line  in  the 
triangle  ?■ 

10.  How  many  pairs  of  opposite  points  are  there 
in  each  form  ? 

11.  Are  there  any  points  in  the  triangle  opposite 
any  other  points  in  it  ? 

12.  The  sum  of  the  angles  of  the  rectangle  is 
equal  to  how  many  right  angles  ? 

13.  The  sum  of  the  angles  of  the  triangle  is  equal 
to  how  many  right  angles  ? 

14:  The  sum  of  the  angles  of  the  triangle  is  equal 
to  what  part  of  the  sum  of  the  angles  of  the  rectangle? 

15.  What  is  there  in  the  triangle  that  is  not  fomid 
in  the  rectano-le? 

IG.  What  is  there  in  the  rectangle  that  is  not 
found  in  the  triangle  I 

17  Without  observing  the  forms  think  of  all  the 
differences  you  can  ? 

18.  Write  the  directions  for  folding  the  square 
into  the  triangle. 

Cutting. — Cut  a  rectangle  2  inches  long  and  1  inch 
wide;  another  3  inches  by  2  inches;  and  another  -t 
inches  by  2  inches.  Pin  them  together.  Do  not 
measure  those  you  bring  to  the  class. 


28  LESSONS    IN    FORM. 

Cut  a  triangle  having  all  its  sides  equal,  another 
having  only  two  sides  equal,  another  of  which  no  two 
sides  are  equal,  and  a  fourth  having  a  right  angle. 
Write  in  the  first,  equilateral  triangle ;  in  the  second, 
isosceles  triangle ;  in  the  third,  scalene  triangle ;  and  in 
the  fourth,  right  triangle. 

Can  you  cut  a  triangle  having  two  right  angles? 

Can  you  cut  a  triangle  so  that  the  sum  of  two 
sides  of  it  shall  be  equal  to  the  third  side  ? 

Drawing. —  Practice  trying  to  draw  rectangles  of 
the  same  dimensions  as  those  you  cut.  Measure  all 
that  you  draw  except  those  that  you  draw  to  bring  to 
the  class. 

Cutting. —  Observe  a  window  at  your  home  and 
cut  a  piece  of  paper  in  the  same  shape,  and  so  that  the 
edges  sliall  have  the  same  relation  to  each  other  as  the 
edges  of  the  window. 

Example  :  If  the  window  is  6  feet  high  and  3  feet 
wide,  cut,  as  nearly  as  you  can,  a  piece  of  paper  6 
inches  long  and  3  inches  wide,  or  6  half  inches  long 
and  3  half  inches  wide,  or  so  that  the  length  of  the 
paper  shall  be  twice  its  width. 

Drawing. —  Observe  the  window  and  draw  it  in 
proper  proportion. 

Measure  the  window  and  write  the  measure  on 
the  paper  on  which  you  made  the  drawing,  stating : 

The  window  is feet  and inches  high^  and 

feet  and inches  wide. 

Give  each  pupil  two  four-inch  squares. 

Have  one  of  the  pupils  give  directions  for  placing 
a  square  and  folding  it  into  a  triangle. 


RECTANGLE   AND   TRIANGLE. 


29 


Unfold  the  paper. 

1.  Do  you  see  the  line  made  by  creasing  the  paper? 

2.  What  does  this  line  connect  ?  The  hne  joining 
the  opposite  points  of  the  square  is  a  diagonal  line. 

3.  What  does  a  diagonal  line  join  ? 

4.  Show  me  opposite  points  of  the  blackboard. 

5.  What  is  a  line  called  which  joins  the  opposite 
corners,  or  points,  of  a  blackboard  ? 

6.  Show  me  what  would  be  a  diagonal  line  of  the 
top  of  your  desk. 

7.  What  is  a  diagonal  line  of  one  of  the  surfaces 
of  a  pane  of  glass?     Of  one  of  the  walls  of  the  room? 

Fold  the  square  again  into  a  triangle.  Place  it  at 
one  side  and  take  the  other  square.  Place  the  square 
so  that  one  edge  shall  be  parallel  to  the  right  edge  of 
your  desk.  Fold  the  square  so  that  the  right-hand 
point  in  front  will  coincide  with  the  left-hand  point  at 
the  back.  Crease  the  paper.  Open  the  paper.  Do 
you  see  the  diagonal  line  if  Fold  the  paper  so  that  the 
front  edo^e  will  coincide  with  the  diao'onal  line.  Crease 
the  paper.  Turn  the  paper  over.  The  form  you  have 
folded  is  a  trapezoid. 


COMPARISON  OF  THE  TRIANGLE  WITH  THE  TRAPEZOID. 


Remakk:     The  important  part  of  the  work  suggested  by  this 
book  consists  in  the  simple  exercises  of  finding  and  expressing  the 


30  LESSORS   i:t^   FORM. 

likenesses  and  differences  of  the  forms  compared,  and  in  finding  sim- 
ilar forms  in  and  outside  of  the  school-room.  These  are  the  exer- 
cises that  will  foster  habits  of  observation,  and  be  of  the  greatest  edu- 
cational value.  At  least  nine-tenths  of  the  time  given  to  the  study 
of  form  should  consist,  not  in  answering  the  questions  of  the  book, 
but  in  discovering  relations  not  suggested  by  the  questions.  Inces- 
sant questioning,  on  the  part  of  teacher  or  text,  fixes  on  the  pupil  the 
habit  of  waiting  to  be  questioned,  and,  when  this  condition  is  induced, 
the  pupil's  thinking  generally  ends  with  the  questioning.  The  teach- 
ers who  will  succeed  in  this  work,  are  those  that  have  the  courage 
to  wait,  and  who  can  make  their  practice  harmonize  with  the  theory 
that  it  is  what  the  pupil  docs  for  himself  that  educates  him. 

1.  Find  likenesses. 

2.  Find  differences. 

3.  Find  trapezoids  in  the  room.* 

4.  Find,  at  least,  five  trapezoids  in  going  to  and- 

from  school  or  at  home. 

Remakk:  Trapezoids  can  be  found  where  building  is  being 
done.     Parts  of  the  roofs  of  many  houses  are  in  this  shape. 

6.  What  are  the  names  of  the  five  objects  which 
you  found  having  triangular  surfaces? 

Write  the  answers  to  the  following  questions: 

1.  Each  form  has,  at  least,  one  pair  of  what  kind 
of  lines? 

2.  The  difference  in  direction  of  two  perpendicu- 
lar lines  is  what  kind  of  an  angle? 

3.  Each  form  has  at  least  one — angle  and  one — 
angle. 

4.  What  is  the  name  of  the  line  opposite  the  right 
angle  in  the  triangle  ? 

5.  The  hypotenuse  of  the  triangle  is  equal  to  what 
line  of  the  trapezoid  ? 

♦Suggestion  :  If  at  any  time  forms  are  being  compared  which  can 
not  be  found  in  the  room,  the  teacher  should  have  several  of  these  forms 
drawn  on  blackboard  for  the  children  to  discover  when  looking  for  forms. 


compauiso:n"  of  thiaxgle  and  trapezoid.        31 

0.  The  sum  of  the  two  equal  hues  of  the  trapezoid 
and  of  the  diagonal  line  is  equal  to  what  in  the  triangle  ? 

7.  To  wiiat  is  the  sum  of  the  two  acute  angles  of 
the  triangle  equal? 

8.  The  sum  of  tlie  angles  of  the  triangle  is  equal 
to  what  in  the  trapezoid  ? 

9.  Can  you  find  two  equal  lines  and  two  equal 
angles  in  the  triangle  ?     Which  are  they  ? 

10.  Can  you  find  two  equal  lines  and  two  equal 
angles  in  the  trapezoid?     Which  are  they  ? 

DIFFERENCES. 

1.  Which  figure  has  the  greater  surface,  or  area? 

2.  How  many  lines  bound  each  form? 

3.  How  many  angles  has  each  ? 

4.  How  many  right,  acute,  and  obtuse  angles  has 
each? 

5.  One  of  the  acute  angles  of  the  triangle  is  equal 
to  what  part  of  one  of  the  right  angles  of  the  trape- 
zoid ? 

6.  The  acute  angle  of  the  trapezoid  is  greater 
than  what  in  the  triangle? 

7.  The  acute  angle  of  the  trapezoid  is  greater  than 
what  part  of  the  right  angle  of  the  triangle  ? 

8.  The  longer  of  the  two  lines  forming  the  obtuse 
angle  of  the  trapezoid  is  shorter  than  what  line  in  the 
triangle,  and  longer  than  what  line? 

9.  An  acute  angle  of  the  triangle  is  less  than  one- 
half  of  what  angle  of  the  trapezoid?  How  do  you 
know  ? 

10.  The  right  angle  in  the  triangle  is  greater  than 
one-half  of  what  angle  in  the  trapezoid  ?     Why  ? 


32  LESSONS    IN    FORM. 

11.  If  a  right  angle  were  equal  to  one-half  of  the 
obtuse  angle  of  the  trapezoid,  to  what,  at  least,  would 
the  obtuse  angle  be  equal  ? 

12.  Do  any  of  the  lines  of  the  triangle  extend  in 
the  same  direction  ?  Any  of  the  lines  of  the  trapezoid  ? 
What,  then,  is  true  of  the  trapezoid  that  is  not  true  of 
the  triangle  ? 

13.  The  sum  of  tne  lines  of  the  triangle  is  less 
than  w^hat? 

14.  The  shortest  line  of  the  trapezoid  is  greater 
than  one-half  of  what,  and  less  than  one-half  of  what, 
in  the  triangle  ? 

15.  The  sum  of  the  two  acute  angles  of  the 
triangle  is  greater  than  what  in  the  trapezoid,  and  less 
than  what  in  it  ? 

16.  "Which  figure  has  opposite  points  and  lines? 
How  many  pairs  of  each  ? 

17.  There  are  no  pairs  of  opposite  points  or  lines 
in  which  figure  ? 

IS.  If  both  pairs  of  opposite  points  of  the  trape- 
zoid were  joined  by  lines,  how  many  diagonal  lines 
would  it  have? 

19.  Are  there  any  diagonal  lines  in  the  triangle? 
Why  not? 

20.  Write  the  direction  for  folding  the  square 
into  a  trapezoid. 

FINDING  FORMS  MADE  BY  FOLDING. 

Give  each  pupil  a  four-inch  square.  Place  it  for 
folding. 

Fold  the  square  so  that  the  right-hand  point  in 
front  will  coincide  with  the  left-hand  point  at  the  back. 

Crease.     Open  the  paper.    Fold  the  square  so  that 


FIXDIXG    FORMS   MADE    BY    FOLDIXG. 


33 


the  left-hand  i)oint  in  front  will  fall  upon  the  right- 
hand  point  at  the  back. 

Crease  Open  the  paper.  Fold  the  paper  so  that 
the  front  edge  will  fall  upon  one  of  the  diagonals. 

Crease.  Fold  the  left-hand  edge  so  that  it  will 
coincide  with  the  same  diagonal. 

Crease.  Fold  the  paper  so  that  the  right-hand 
point  at  the  back  will  fall  upon  a  diagonal,  and  so  that 
an  isosceles  triangle  will  be  formed. 

Crease  paper  carefully.  Unfold  the  paper  so  that 
you  will  have  the  square  again.  Observe  the  forms 
made  by  the  creased  lines. 


1.  Observe  the  figure.    How  many  triangles,  each 
having  a  right  angle,  can  you  find? 

2.  How  many  triangles  having  two  sides  equal 
can  yuu  find  { 

3.  How  many  triangles  canyon  lind  having  no 
two  lines  equal. 

4.  How  many  trapezoids  can  you  InuH 


34  LESSON'S    IN"    FORM. 

5.  How  many  different  figures  have  you  found  in 
the  square  ? 

Give  each  pupil  two  four-inch  squares.  Have  some 
pupil  give  direction  for  placing  the  square  and  folding 
it  into  a  trapezoid. 

Take  the  other  square.  Fold  it  so  that  the  right- 
hand  point  in  front  will  fall  upon  or  coincide  with  the 
left-hand  point  at  the  back.  Crease  the  paper.  Open 
it.  Fold  the  paper  so  that  the  edge  in  front  will  coin- 
cide with  the  diagonal.  Crease.  Fold  the  paper  so 
that  the  edge  at  the  back  will  coincide  with  the  diag- 
onal. Crease.  Turn  the  paper  over.  This  form  is 
called  a  rhomboid. 

OBLIQUE  ANGLES  AND  LINES. 
Obtuse  and  acute  angles  are  called  oblique  angles. 

1.  How  many  oblique  angles  are  there  in  the 
rhomboid?     Find  oblique  angles  in  the  room. 

2.  What  angles  are  called  oblique  angles? 

3.  Are  there  any  oblique  angles  in  the  square  ? 

4.  Are  there  any  oblique  angles  in  the  trapezoid? 
How  many  ? 

5.  The  lines  forming  either  acute  angles  or  obtuse 
angles  are  called  oblique  lines.  How  many  pairs  of 
oblique  lines  are  there  in  the  rhomboid  ? 

6.  Find  oblique  lines  in  the  room.  Are  any  of 
the  lines  of  a  rectangle  oblique  lines  ? 

7.  "What  kind  of  angles  are  formed  by  oblique 
lines  ? 


COMPARISON    OF   TUAPEZOID    WITH    RHOMBOID. 


:^5 


COMPARISON   OF   THE    TRAPEZOID    WITH   THE  RHOM- 

BOID. 


1.  Find  likenesses. 

2.  Find  differences. 

3.  Find  rhomboids  in  room. 

4.  Find,  at  least,  five  rhomboids,  and  report  the 
names  of  the  objects  that  have  surfaces  of  this  shape. 

5.  What  objects  did  you  find  having  a  surface  or 
surfaces  in  the  shape  of  a  trapezoid  ? 

QUESTIONS— LIKENESSES. 

Write  answers. 

1.  Each  surface  is  bounded  by  how  many  lines 
and  has  how  many  angles  ? 

2.  How  many  pairs  of  opposite  points  and  lines 
are  there  in  each  ? 

3.  How  many  pairs  of  adjacent  lines  has  each  ? 

4.  How  many  diagonal  lines  can  each  have  ? 

6.  The  longer  diagonal  of  the  rhomboid  is  equal 
to  what  in  the  trapezoid  ? 

6.  The  obtuse  angle  of  the  trapezoid  is  equal  to 
wliat  in  the  rhomboid  ? 

Y.  One  of  tlie  acute  angles  in  the  rhoml)oid  is 
c(jual  to  wliat  in  tlie  trapezoid  ? 

8.  Tlie  sliortest  lines  of  the  trapezoi:!  is  equal  to 
what  in  the  rhomboid  'i 


36       •  LESSORS  i:n"  form. 

9.  The  longest  boundary  line  of  the  trapezoid  is 
equal  to  what  in  the  rhomboid  ? 

10.  The  sum,  of  the  longer  diagonal  line,  the 
shortest  line  and  the  longest  line  of  the  trapezoid  is 
equal  to  what  in  the  rhomboid  ? 

11.  There  is,  at  least,  one  pair  of  opposite  lines 
extending  how  in  each  form  ? 

12.  That  part  of  the  trapezoid  bounded  by  its 
longest  line,  shortest  line  and  the  longer  diagonal  line 
is  equal  to  what  in  the  rhomboid  ? 

13.  Can  you  find  two  pqual  angles  and  two  equal 
lines  in  each  ?     What  or  which  are  they  ? 

DIFFERENCES. 

1.  How  man}^  different  angles  are  there  in  each 
form  ? 

2.  What  kind  of  lines  are  found  in  the  trapezoid 
that  are  not  found  in  the  rhomboid  ? 

3.  How  many  pairs  of  parallel  lines  are  there  in 
each  ? 

4.  One  of  the  two  equal  lines  of  the  trapezoid  is 
longer  than  what  and  shorter  than  what  in  the  rhom- 
boid ? 

5.  The  shorter  diagonal  of  the  trapezoid  is  longer 
than  what  in  the  rhomboid  and  shorter  than  what  in 
the  same  figure  ? 

6.  Which  form  has  the  greater  area? 

7.  Write  the  directions  for  folding  the  rhom- 
boid. 

Cutting. — Observe  and  cut  two  straight  line  fig- 
ures in  proper  pro])ortion,  such  as  doors,  windows, 
floors,  walls,  blackboards,  tops  of  tables,  sides  and  ends 


COMPAiaSOX    Ob'    i:iI().M!5Ull)    Wnu    TRAPEZIUM. 


37 


(.A  boxes,  covers  of  books,  etc.,  etc.  Measure  the  two 
dimensions  of  the  objects  observed  and  write  the  meas- 
ure on  the  paper  cut.  - 

Dkawixg. — Observe  and  draw  two  straight  line 
figures.  Write  the  measures  of  the  two  dimem-ions  on 
the  drawing. 

Can  you  (h'aw  a  trapezoid,  having  only  one  right 
anoie  ? 

Give  eacli  pupil  two  squares. 

Have  some  pupil  give  directions  for  placing  and 
folding  a  square  into  a  rhomboid. 

Directions  for  folding  the  square  into  a  trapezium  : 

Place  a  square  so  that  its  right  edge  will  extend  in 
the  same  direction  as  the  right  edge  of  the  desk. 

Fold  the  square  so  that  the  right  point  in  front  will 
fall  upon  the  left  point  at  the  back.     Crease. 

Open  the  paper.  Fold  the  paper  again  so  that  the 
front  edge  will  lie  along  the  diagonal  line.  Crease. 
Fold  the  pa])er  so  that  the  left  edge  will  lie  along  the 
same  diagonal.  Crease.  Turn  the  paper  over  and 
place  it  for  comparison  with  the  rhomboid. 

COMPARISON   OF  THE   RHOMBOID   WITH   THE  TRA- 
PEZIUM. 


1.     Find  likenesses. 


38  LESSONS   IK   rORM. 

2.  Find  differences. 

3.  Find  trapeziums  in  room. 

4.  Find  five  objects  that  have  surfaces  or  a  surface 
in  the  shape  of  a  trapezium.  Any  four-sided  figuro 
having  none  of  its  edges  parallel  is  a  trapeziura. 

5.  What  objects  did  you  find  having  a  surface  or 
surfaces  in  the  shape  of  a  rhomboid? 

QUESTIONS— LIKENESSES. 
Write  answers  to  the  following  questions: 

1.  The  two  long  lines  of  the  trapezium  are  equal 
to  what  in  the  rhomboid  ? 

2.  The  two  short  lines  of  the  trapezium  are  equal 
to  what? 

3.  What  is  true  of  one  of  the  diagonals  of  each  9 

4.  Into  what  does  the  longer  diagonal  of  each 
figure  divide  it? 

5.  The  area  of  half  of  the  trapezium  is  equal  to 
the  area  of  what  ? 

6.  The  area  of  the  rhomboid  equals  what  ? 

7.  Each  form  has  how  ma;ny  obtuse  angles  ? 

8.  The  sum  of  the  right  angle  and  the  acute  angle 
of  the  trapezium  equals  what  in  the  rhomboid? 

9.  If  the  two  obtuse  angles  of  each  form  are 
equal,  and  if  the  sum  of  the  right  angle  and  acute  angle 
of  the  trapezium  equal  the  sum  of  the  two  acute  angles 
of  the  rhomboid,  to  what  is  the  sum  of  the  angles  of 
the  trapezium  equal  ? 

DIFFERENCES. 
1.     What  is  true  of  two  adjacent  lines  of  the  tra- 


CLASSIFICATION   OF    FORMS. 


89 


peziiim  that  is  not  true  of  any  of  the  adjacent  lines  of 
the  rhomboid? 

2.  AVhat  is  true  of  the  two  short  Hnes  of  the  tra- 
pezium that  is  not  true  of  any  of  the  lines  of  the  rhom- 
boid ? 

3.  What  kind  of  an  angle  is  found  in  the  trapezium 
that  is  not  found  in  the  rhomboid? 

4.  What  is  true  of  the  opposite  lines  of  the  rhom- 
boid that  is  not  true  of  the  opposite  lines  of  the  trape- 
zium ? 

5.  Are  all  trapeziums  kite-shaped  ? 

6.  Must  a  trapezium  have  a  right  angle  ? 

7.  Can  a  trapezium  have  two  right  angles  ? 

8.  Can  a  trapezium  have  two  adjacent  right 
angles  ? 

9.  Write  the  direction  for  folding  a  square  into  a 
trapezium. 

CLASSIFICATION  OF  FORMS. 


Quadri-lateral , 

Parallelogram , 

Kectangle , 


Quadrilateral , 

rarallelogram , 

Rectangle, 


Quadrilateral , 
Parallelogram , 


a 


Quadrilateral, 
Trapezoid . 


In  wliat  respects  are  all  these  iigurcs  alike? 


40  LESSONS   IN   I'OUM. 

2.  What  general  name  can  be  used  in  speaking  of 
any  of  the  above  forms  ? 

3.  In  what  are  a,  h  and  e  ahke  ? 

4.  By  what  common  name  can  you  speak  of  a,  h 
and  c? 

5.  What  common  name  hav^e  a  and  h  ? 

6.  Are  all  quadrilaterals  parallelograms  ?  Are  all 
parallelograms  quadrilaterals  ? 

7.  Are  all  rectangles  parallelograms?  Are  all 
parallelograms  rectangles  ? 

8.  Are  all  squares  rectangles?  Are  all  rectangles 
squares  ?     Why  not  ? 

9.  What  is  true  of  an  oblono:  rectano^le  that  is  not 
true  of  the  square?  What  is  true  of  a  square  that  is 
not  true  of  an  oblong  rectangle  ? 

10.  In  what  respects  are  the  square,  oblong  rect- 
angle and  trapezoid  alike  ? 

11.  How  many  pairs  of  parallel  lines  has  a  square, 
an  oblong  rectangle,  and  a  trapezoid,  respectiveljT'  ? 

12.  A  parallelogram  is  a  quadrilateral  whose 
opposite  sides  extend  in  the  same  direction,  or  are  par- 
allel. 

13.  Why  is  a  square  a  parallelogram  ? 

14.  Is  a  rhomboid  a  parallelogram  ?     Why  ? 

15.  Canyon  draw  a  quadrilateral  that  is  not  a 
parallelogram  ? 

16.  Can  you  draw  a  quadrilateral  that  is  neither 
a  parallelogram  nor  a  trapezium  ?     W^hat  is  its  name? 

17.  Is  a  trapezoid  a  quadrilateral  ?  Is  a  trapezoid 
a  parallelogram ?  Are  both  pairs  of  opposite  sides  in 
the  trapezoid  ]mrallel  ? 

18.  Can   you   find  five  angles,  or  differences  in 


coMPArasox  of  scalene  triangle  and  tiial'ezh  m.  41 

direction,  in  the  trapezoid?      Can   you  iiiul   six  in  a 
trapezium? 

19.  Draw  a  figure  having  at  least  one  right  angle 
and  four  equal  sides.     What  is  its  name? 

20.  Draw  a  figure  liavin^:  at  least  one  pair  of 
perpendicular  lines  and  four  equal  sides.     What  is  it? 

TRAPEZIUM  AND   SCALENE   TPtlANGLE. 

Give  each  pupil  two  squares.  Have  some  pupil 
give  the  directions  for  folding  the  trapezium.  Place 
the  trapezium  at  one  side,  and  have  another  pupil 
give  the  directions  for  folding  the  other  square  into 
a  trapezium. 

Observe  the  trapezium.  Do  you  see  an  obtuse 
angle?  Do  you  see  the  point  a,t  which  the  two  lines 
forming  the  obtuse  angle  meet?  This  point  is  the 
vertex  of  the  angle.  How  many  angles  has  the  trape- 
zium ?     How  many  vertices  ? 

Directions  for  folding  tlie  trapezium  into  a  scalene 
triangle : 

Fold  the  trapezium  so  that  the  vertex  of  one  of  the 
obtuse  angles  will  fall  upon  the  vertex  of  the  other 
obtuse  angle.  Crease  the  paper.  Write  a  comparison 
of  the  scalene  triangle  with  the  trapezium. 


1.     Give  likenesses. 


42  LESSOKS   11^   FORM. 

2.  Give  differences. 

3.  Find  scalene  triangles  in  the  room. 

4.  Write  the  names  of  five  objects  that  have  sur- 
faces in  the  shape  of  trapeziums. 

5.  Find  five  scalene  triangles  at  home  or  in  going 
to  and  from  school. 

6.  Write  the  directions  for  folding  the  square 
into  the  scalene  triangle. 

Have  pupils  cut  and  bring  to  the  class  to-morrow 
one  of  each  of  the  forms  compared.  Have  them  cut 
trapezoids  and  trapeziums  differing  in  shape  from  the 
ones  that  have  been  compared.  When  the  forms  are 
brought  in,  mix  them  and  have  pupils  give  name  and 
description  of  the  form  selected.  Example :  This  quad- 
rilateral is  a  trapezoid  as  it  has  only  two  sides  parallel. 
This  quadrilateral  is  a  rectangle  as  its  angles  are  right 
angles.  It  is  an  oblong  rectangle,  because  it  is  longer 
than  it  is  wide. 

Have  pupils  find  forms  in  the  collection,  from 
descriptions  given  by  other  pupils.  Have  just  enough  of 
a  description  given  to  enable  the  one  who  selects  to 
find  the  form.  Let  the  pupil  w^ho  gives  the  descrip- 
tion name  the  one  who  is  to  find  the  form.  Ex- 
am>ples : 

1.  Find  a  quadrilateral  having  two  pairs  of  par- 
allel lines.  What  is  its  name  ?  Find  another  liaving  a 
different  name  but  having  two  pairs  of  parallel  lines. 
What  is  its  name  ? 

2.  Find  a  quadrilateral  having  only  tAvo  lines 
parallel.     What  is  its  name  ? 


COMPATHSON"  OF  SCALENE  TRIAN-QLE  AND  RHOMBUS.     43 

3.  Find  a  parallelogram  of  which  any  two  of  its 
adjacent  lines  are  equal      Wliat  is  its  name? 

4.  Find  a  rectangle  that  is  longer  than  it  is  wide. 
What  is  its  name? 

Remaiik:  These  exercises  will  lea(\  to  close  observation, 
which  is  the  basis  of  correct  expression  and  exact  reasoning, 

THE  SCALENE  TRIANGLE   AND   RHOMBUS. 

Give  pupils  two  squares. 

Have  a  pu[)il  give  the  direction  for  folding  a  square 
into  a  scalene  triangle. 

Have  another  pupil  give  the  direction  for  folding 
a  square  into  a  trapezium. 

Direction  for  folding  the  trapezium  into  a  rhom- 
bus :  Fold  the  trapezium  so  that  one  of  its  short  lines 
will  coincide  with  the  diagonal.  Crease.  Fold  the 
paper  so  that  the  other  short  line  will  fall  upon  the 
diagonal.     Crease. 

Write  a  comparison  of  the  scalene  triangle  with 
the  rhombus : 


1.  Give  likenesses. 

2.  Give  differences. 

3.  Find  rhombuses  in  room. 

4.  What  objects  did  you  find  having  a  scalene 
triangle  for  a  surface  ? 


44 


LESSONS    IN   FORM. 


5,  Find  live  objects  that  have  a  surfacG  or  sur- 
faces that  are  rhombuses,  and  give  the  names  of  the 
objects  to-morrow. 

6.  Write  the  directions  for  fokling  the  rhombus. 

FINDING  FORMS  MADE  BY  FOLDING. 

Give  each  pupil  a  square. 

Fold  the  square  so  that  the  right-hand  point  in 
front  will  fall  upon  the  left-hand  point  at  the  back. 

Crease.  Open  the  paper.  Fold  the  paper  so  that 
the  left-hand  point  in  front  will  fall  upon  the  right-hand 
point  at  the  back. 

Crease  and  open  the  paper.  Fold  each  point  to 
the  center  and  crease.     Open  the  paper. 


How  many  squares,  oblong  rectangles,  trapezoids 
and  triangles  can  you  find  in  the  figure  ? 

Draw  each  fio:ure.  Trv  to  draw  the  fificures  the 
same  size,  and  in  the  same  proportion,  as  those  you  see 
in  the  square. 


RHOMBUS   AND    ISOSCELES   TRIANGLE. 


45 


THE  RHOMBUS  AND  THE  ISOSCELES  TRIANGLE. 
Give  each  pupil  tAvo  scpiares. 

Have  pupils  give  directions  for  folding  each  square 
into  a  rhombus. 

The  following  are  the  directions  for  folding  the 
rhombus  into  an  isosceles  triangle  : 

Fold  the  rhombus  so  that  the  vertex  of  one  of  the 
acute  angles  will  fall  upon  the  vertex  of  the  otlier 
acute  angle.     Crease  the  paper. 

Write  a  comparison  of  the  rhombus  with  the 
isosceles  triangle. 


1.  Find  likenesses. 

2.  Find  differences. 

3.  Find  isosceles  triangles  in  room. 

4.  Write   the  names   of  the  objects  which  you 
found  having  a  surface  or  surfaces  like  a  rhombus. 

5.  Find  five  objects  which  have  a  surface  in  the 
shape  of  an  isosceles  triangle. 

C).     Write  the  direction  for  folding  the  isosceles 
triangle. 


46 


LESSONS   IN"   FORM. 


CLASSIFICATION   OF  FORMS. 

Suggestion:     Omit  work  on  classification  of  forms  uniil  third 


year. 


Quadrilateral , 
Trapezoid , 


Quadrilateral , 
Parallelosram, 
Rhomboid. 


uadrilatcral, 
Parallelogram, 
Rhomboid, 
Rhombus. 


Quadrilateral , 

Paral  1  e  1  ogram , 

Rectangle. 


Quadrilateral, 

Parallelogram, 

Rectangle, 

Square . 


Write  such   a  description  of  each   of  the   above 
forms  that  the  person  reading  it  can  select  the  form 

described,     l^.Iake  the  descriptions  as  short  as  you  can. 

Example :     It  is  a  quadrilateral  with  no  sides  parallel. 

What  is  the  name  of  the  figure  described  ? 

1.  What  is  wrong  in  the  following  description  of 
a  square  ? 

A  square  is  a  figure  bounded  by  four  equal  lines. 
Of  how  many  of  the  forms  is  the  description  true? 

2.  What  is  wrong? 

A  square  is  a  figure  bounded  by  four  lines  and 
having  four  right  angles. 


CLASSIFICATIONS'   OF   FORMS.  47 

Of  how  many  figures  is  the  description  true? 

3.  What  is  wrong? 

A  rhombus  is  an  oljlique-angled  parallelogram. 
How  many  figures  does  this  describe  ? 

4.  What  is  wrong  ? 

A  rhombus  is  a  figure  having  four  equal  lines. 
How  many  figures  are  included  in  this  description? 

5.  A  quadrilateral  is  a  plane  figure  bounded  by 
four  straight  lines. 

How  many  of  the  above  figures  are  quadrilaterals? 

6.  A  quadrilateral  whose  opposite  lines  are  paral- 
lel is  a  parallelogram. 

How  many  of  the  figures  are  parallelograms? 

7.  A  rhomboid  is  a  parallelogram  whose  angles 
are  oblique  angles. 

How  many  of  the  figures  are  rhomboids? 
Is  a  rhombus  a  rhomboid  ? 
Are  all  rhomboids  rhombuses  ? 

8.  A  rectangle  is  a  parallelogram  whose  angles 
are  right  angles. 

How  many  of  the  figures  are  rectangles? 

Is  a  square  a  rectangle  ? 

Are  all  squares  rectangles  ?  Are  all  rectangles 
squares  ? 

Call  attention  to  the  different  figures  cut  out  of 
paper,  drawn  on  the  blackboard,  or  found  in  the  room, 
and  have  pupils  tell  what  they  are  and  define  them.  Ex- 
amjjle:  The  blackboard  is  a  rectangular  parallelogi-am 
because  its  opposite  lines  are  parallel  and  its  angles  are 
ri<iiit  angles. 


48 


lesso:n^s  ij^  form. 


Suggestion:  It  would  be  well  to  have  the  pupils  re-write  the 
comparisons,  condensing  them  as  much  as  they  can.  In  comparing 
the  square  with  the  rectangle,  all  the  likenesses  could  be  given  in  one 
or  two  sentences,  as,  each  form  is  a  surface  having  four  lines,  four 
points,  four  pairs  of  perpendicular  lines,  four  pairs  of  adjacent  lines, 
two  pairs  of  opposite  lines,  two  pairs  of  pirallel  lines,  and  four  right 
angles . 

THE  ISOSCELES  TRIANGLE  AND  THE  HEXAGON. 

Give  each  pupil  two  4-incli  squares.  Have  a  pupil 
give  the  directions  for  folding  the  square  into  an  isos- 
celes triangle.  Have  another  pupil  give  the  directions 
for  folding  the  other  square  into  a  rhombus. 

The  following  are  the  directions  for  folding  the 
rhombus  into  a  hexagon :  Fold  the  rhombus  so  that 
the  vertex  of  one  of  the  acute  angles  will  fall  upon  the 
center  of  the  rhombus.  Crease.  Fold  the  paper  so  that 
the  vertex  of  the  other  acute  angle  will  coincide  with 
the  same  point.  Crease.  Compare  the  isosceles  triangle 
with  the  hexao-on.     Write. 


1.  Find  likenesses. 

2.  Find  differences. 

3.  Find  hexagons  in  room. 

4.  Write  the  names  of  the  objects  in  which  you 
found  isosceles  triangles. 


CUTTING    AND    DRAWING. 


49 


CUTTING  AND  DRAWING. 

Have  pupils  observe  and  cut  in  outline  forms  in  art 
and  of  animals,  inkstands,  flower-pots,  buckets,  baskets, 
ladders,  cups,  teapots,  cows,  horses,  pigs,  etc.,  etc.  Pre- 
serve pupils'  work  by  pasting  it  on  cardboard. 

Have  pupils  observe  and  draw  in  outline  the  same 
forms  that  they  cut. 

The  following  are  a  few  examples  of  the  many 
things  that  may  be  observed  and  cut  in  outline  and 
observed  and  drawn.  The  things  themselves,  and  not 
these  diagrams,  are  to  be  observed  in  the  cutting  and 
drawing. 


50 


LESSOI^S   IN   FORM. 


]£ 


w w 


[SI 


q — 3 


POINTS   AND   LINES. 


51 


POINTS  AND  LINES. 


QUESTIONS. 

Suggestion:  Omit  work  on  points,  lines  and  surfaces  with 
first,  second,  and  third  year  pupils.  Omit  with  fourth  year  pupils 
until  after  the  comparisons  of  the  solids. 

1.  If  a  point  be  moved,  its  path  is  what  ?  What, 
then,  is  the  path  of  a  moving  point  ? 

2.  The  path  of  a  point  that  does  not  change  its 
direction  is  what  kind  of  a  hne?  What,  then,  is  a 
sti'aight  line  ? 

3.  If  a  point  is  moved  so  as  to  change  its  direction 
continuall}^,  what  kind  of  a  hne  will  it  make,  or  gen- 
erate?    What,  then,  is  a  curved  line? 

4.  If  two  points  are  moved  in  the  same  direction, 
wliab  will  be  true  of  the  lines  generated?  How  may 
parallel  lines  be  generated? 

5.  How  far  must  a  point  be  moved  to  generate  a 
line? 

1.  Does  a  point  have  length  ? 

2.  Would  two  or  more  points  have  length  ? 

3.  What  dimension  does  a  line  have  ? 


52  LESSONS   TIT   FORM. 

4.  Is  a  part  of  a  line  a  line  ? 

5.  Is  a  point  a  part  of  a  line  ?     "VYhy  not? 

6.  Would  a  number  of  points  make,  or  constitute, 
a  line?     Why  not? 

7.  If  you  know  the  position  of  two  points  in  a 
straight  line,  can  you  tell  the  position  and  direction  of 
the  line? 

POINTS,  LINES  AND  SURFACES. 

QUESTIONS. 

1.  What  will  a  moving  line  make,  or  generate  ? 

2.  How  can  a  line  be  moved  so  as  not  to  make,  or 
generate,  a  surface  ? 

3.  When  a  line  is  moved  so  as  to  generate  a  sur- 
face, what  will  the  extremities  of  the  line  generate  ? 

4.  How  far  must  a  line  be  moved  to  generate  a 
surface  ? 

5.  If  a  straight  line  is  moved  so  as  to  continue 
parallel  to  its  first  position,  what  is  true  of  the  length 
of  the  lines  that  its  extremities  generate?  What  is 
true  of  the  direction  of  the  two  lines  generated  by  its 
extremities  ? 

6.  What  two  thing  are  true  of  the  lines  gener- 
ated by  the  extremities  of  a  line  moved  so  as  to  con- 
tinue parallel  to  its  first  position  ? 

7.  If  a  straight  line  is  moved  so  that  its  extremi- 
ties generate  curved  lines,  what  will  the  line  itself  gen- 
erate ?  How  must  a  line  be  moved  so  as  to  generate  a 
curved  surface? 

8.  Could  a  curved  surface  be  generated  by  a 
straight  line,  if  one  of  its  extremities  did  not  change  its 
position? 


POINTS,    LINES   AND    SURFACES.  53 

9.  If  a  straight  line  is  moved  so  that  its  extremi- 
cies  generate  straight  lines,  what  kind  of  a  surface  does 
tb<8  line  itself  generate? 

10.  What  is  the  name  of  the  figure  generated  by 
moving  a  straight  line  so  that  its  extremities  generate 
equal  straight  lines  ? 

11.  How  may  a  parallelogram  be  generated? 

12.  If  a  straight  line  is  moved  so  that  its  extremi- 
ties generate  equal  straight  lines,  which  are  perpendic- 
ular to  the  given  line  in  its  first  position,  what  figure  is 
generated  by  the  line  itself? 

13.  How  may  a  rectangle  oe  generated? 

14.  How  may  a  square  be  generated? 

15.  How^  may  a  square  be  generated,  beginning 
with  a  point  ? 

16.  AVill  the-extremities  of  -a  straight  fine  gener- 
ate straight  lines  unless  both  extremities  are  moved  at 
the  same  rate  ? 

IT.  Can  a  trapezoid  be  generated  by  a  straight 
line? 

1.  How^  many  dimensions  has  a  line? 

2.  How  many  dimensions  has  a  surface  ? 

3.  How  many  dimensions  has  a  part  of  a  sur- 
face ? 

4.  Is  a  line  a  part  of  a  surface? 

5.  Has  a  line  width?  Would  two  or  more  lines 
have  wadth  ?  Would  a  number  of  lines  make,  or  con- 
stitute a  surface  ? 


Part  Second 


SOLIDS. 

THE   SPHERE. 

Have  pupils  make  a  sphere  of  clay.  If  the  clay  is 
not  at  hand,  place  a  sphere,  or  ball  of  some  kind,  before 
them  for  observation. 


1.  Have  you  seen  other  forms  of  this  shape  in  or 
about  the  school-room  ? 

2.  What  are  the  names  of  the  objects  that  3^ou 
have  observed  in  the  shape  of  a  sphere  ? 

3.  Name  five  objects  that  you  have  seen  at  home 
or  in  other  places,  in  the  shape  of  a  sphere.  Example: 
I  have  seen  some  grapes  m  the  shape  of  a  sphere. 

54 


OBLATE    SPHEROID.  55 

4.  The  sphere  is  limited,  or  bounded,  by  what 
kind  of  surface  ? 

5.  Find  objects  in  the  room  that  are  limited  or 

partially  limited  by  a  curved  surface,  and. tell  the  name 
of  each. 

6.  Hand  me,  to-morrow,  the  names  of  ten  objects 
that  you  have  observed  that  have  a  curved  surface. 

Drawing. —  1.  Observe  and  draw  some  object  in 
the  si] ape  of  a  sphere. 

2.  Draw,  from  memory,  another  sphere  two  inches 
in  diameter.     Measure. 

AN  OBLATE  SPHEROID. 


1.  Observe  and  model  out  of  clay  an  oblate 
spheroid."^ 

2.  Have  you  seen  any  obiate  spheroids  at  school? 
Example:     Some  door-knobs  are  oblate  spheroids. 

3.  What  are  the  names  of  five  oblate  spheroids 
that  you  have  seen  at  home  or  in  other  places  ? 

4.  To-morrow,  hand  me,  on  paper,  the  names  of 
five  oblate  spheroids. 

Drawing. —  1.  Observe  and  draw  an  oblate 
spheroid. 

*  Note  :  If  you  do  uotluive  an  oblutc  spheroid, uso  ii  door-knol),  a  field 
turnip,  or  some  other  common  object  in  tlie  shape  of  un  oblate  spheroid,  for 
a  model. 


66  LESSONS   IN   FOKM. 

2.  Draw  another  oblate  spheroid  with  one  of  its 
longest  diameters  two  inches,  and  its  shortest,  one  inch. 
Measure. 

A  PROLATE   SPHEROID. 


1.  Observe  and  make  out  of  clay  a  prolate 
spheroid.  A  potato  in  the  shape  of  a  prolate  spheroid 
Avill  do  for  a  model. 

2.  What  objects  have  you  seen  in  the  shape  of  a 
prolate  spheroid. 

3.  Hand  in,  to-morrow,  on  papei',  the  names  of 
five  prolaie  spheroids. 

4.  Does  the  entire  surface  of  a  prolate  spheroid 
curve  uniformly  ? 

5.  AVhat  can  3'ou  say  of  a  curved  surface  of  the 
sphere  ? 

Dra^wing. — Observe  and  draw  some  object  in  the 
shape  of  a  prolate  spheroid. 

CONVEX  AND  CONCAVE. 

1.  The  surface  of  a  drop  of  water  is  convex. 
Find  five  convex  sui^faces  in  the  room. 

2.  Recall  five  objects  that  have  convex  surfaces. 

3.  Observe  objects  and  write  five  sentences  using 
in  each  the  term  convex. 

4.  A  surface  tliat  is  hollow  and  curved  or  rounded 


CONVEX   AND    CONCAVE.  .     57 

is  concave.     Example :     The  hollow  of  the  hand  or  the 
inside  surface  of  a  watch  crystal  is  concave. 

5.  Find  five  concave  surfaces  in  the  room. 

6.  Observe  objects  and  write  five  sentences  using 
in  eacli  the  term  concave. 

7.  Write  the  names  of  five  objects  that  have 
both  concave  and  convex  surfaces. 


^*  curved  Line 

1.  Find  five  curved  lines,  or  edges,  in  the  room. 

2.  Eecall  five  objects  that  you  have  seen  that 
have  curved  edges,  or  lines.  Example :  The  edge  of 
the  tire  of  a  wagon-wheel  is  curved. 


CoMPARisojsr  OF  Solids.* 

1.  Have  each  pupil  make  a  sphere  of  clay. 

2.  Make  another  sphere  of  the  same  size  as  the 
first.  Use  a  small  wire  and  cut  one  of  the  spheres  into 
hemispheres. 


*IlEMAiiK:  If  there  is  no  clay  provided  for  niakinjj;- the  solids  to  be 
used  in  the  comparisons,  f^et  forms  made  of  wood  and  use  them.  Do  not 
use  drawings  as  tlic  basis  for  the  comparisons.  Drawinjrs  are  poor  substi- 
tutes for  concrete  represcnt^itions  of  the  forms  to  bo  compared. 


\ 


u  if  i  V  :- 


58  LESSONS   IK   FORM. 

COMPARISON  OF  THE  SPHERE  WITH  THE  HEMISPHERE. 


1.  Find  likenesses. 

2.  Find  differences. 

3.  Recall  objects  in  the  shape  of  a  hemisphere, 
Examjple :     Some  birds's  nest  are  like  a  hemisphere. 

DRAWING. 

Cut  out  of  pasteboard  a  circle  having  a  diameter 
of  1  foot. 

1.  Place  the  circle  on  some  object  whose  top  is 
on  a  level  witli  the  eyes  of  the  pupils,  so  that  the  edge 
of  the  circle  Avill  appear  to  be  a  horizontal  straight 
line.  Direct  pupils  to  observe  the  circle  and  draw  just 
what  they  see. 

2.  Place  the  circle  so  that  its  edge  will  appear  to 
be  a  vertical,  or  upright,  hne  to  the  pupils  in  the  middle 
row.  Direct  pupils  to  observe  the  circle  and  represent 
what  they  see. 

Remark  :  If  pupils  draw  what  they  see,  the  middle  row  will 
draw  straight  lines  only.  Each  of  the  other  pupils  will  draw  an 
ellipse.  The  pupils  who  will  see  an  ellipse  approaching  nearest  the 
circle  will  be  those  in  the  front  seat  of  the  right  and  left-hand  rows. 


COMPARISO^^"   OF  SPHERE  WITH    HEMISPHERE.  59 

3.  Place  the  circle  so  that  the  right  or  the  left- 
hand  row  will  see  the  entire  circle.  Direct  pupils  to 
draw  what  the 7  see. 

Remark  :  If  pupils  are  in  the  habit  of  determining  the  pro- 
portion of  figures  by  holding  the  pencil  at  arm's  lengtli  and  measur- 
ing with  the  eye,  it  would  be  well  to  make  the  measurements  after 
and  not  before  the  drawing  is  made. 

Cut  out  of  pasteboard  a  large  square  and  other 
plane  figures,  and  place  them  in  different  positions 
before  the  class  for  drawing.  If  pupils  can  represent 
just  what  they  see  of  plane  figures,  it  will  aid  them 
greatly  in  drawing  solids  correctly. 

DraAV  a  sphere  and  a  hemisphere. 

1.  Write  likenesses. 

2.  Write  differences. 

QUESTIONS  ON  THE  Sf^HERE  AND  HEMISPHERE. 

LIKENESSES. 

1.  Each  form  has  what  kind  of  a  surface? 

2.  What  kind  of  a  curved  surface  does  each  have  ? 

3.  What  is  true  of  all  the  lines  extending  from 
the  center  of  the  sphere  to  its  surface?  One  of  these 
lines  is  called  a  radius;  two  or  more  of  them  radii. 
What  is  true  of  the  radii  of  the  sphere? 

4.  Make  .a  point  in  the  center  of  the  plane  sur- 
face of  the  hemisphere.  What  is  true  of  all  the 
straight  lines  extending  from  the  center  of  the  plane 
surface  of  the  hemisphere  to  its  curved  surface? 

5.  Wliat  lines  are  equal  in  both  the  sphere  and 
hemisphere? 

G.  AVliat  is  true  of  the  center  of  the  sphere  and 
of  the  center  of  the  plane  surface  of  the  hemisphere  ? 


60  LESSONS   IN"   FORM. 

Y.  In  what  respect  are  the  circumference  of  the 
plane  surface  of  the  hemisphere  and  the  curved  sur- 
face of  'the  sphere  alike?  Do  they  each  curve  uni- 
formly ? 

8.  A  straight  line  drawn  from  the  center  of  the 
plane  surface  of  the  hemisphere  to  its  circumference  is 
called  a  radius.  What  is  true  of  the  radii  of  the  circle 
limiting  the  hemisphere  ? 

9.  How  does  a  radius  of  the  circle  compare  with 
a  radius  of  the  sphere  ? 

10.  Straight  lines  passing  through  the  center  of 
the  sphere  and  terminating  in  its  surface  are  called 
diameters  of  the  sphere,  and  straight  lines  passing 
through  the  center  of  the  plane  surface  of  the  hemi- 
sphere and  terminating  in  its  circumference  are 
diameters  of  the  circle.  What  is  true  of  the  diameters 
of  the  sphere  and  of  the  plane  surface  of  the  hemi- 
sphere ? 

DIFFERENCES. 

1.  Each  solid  is  limited  by  how  many  surfaces? 

2.  The  hemisphere  is  limited  by  what  kind  of  a 
surface  that  is  not  found  in  the  sphere? 

3.  What  is  the  name  of  the  plane  surface  of  the 
hemisphere  ? 

4.  Find  other  circles  in  the  room. 

5.  Kecall  five  objects  that  are  limited  by  circles. 
Example :     The  bottoms  of  some  buckets  are  circles. 

6.  What  kind  of  a  line  did  you  find  in  the  hemi- 
sphere ? 

7.  A  hemisphere   is   equal   to   what  part  of  a 
sphere  ? 


HEMISPHEEE    AND    QUADRANT    OF   A    SPHERE. 


61 


8.  The  curved  surface  of  the  hemisphere  is  equal 
to  what  part  of  the  curved  surface  of  the  sphere  ? 

9.  Can  you  draw  a  straight  line  on  the  surface 
of  each  solid  ? 

10.  Can  you  draw  a  curved  line  on  the  plane  sur- 
face of  the  hemisphere? 

11.  What  two  things  are  found  in  the  hemisphere 
that  are  not  found  in  the  sphere? 

12.  Describe  a  sphere,  without  naming  it,  so  that, 
by  the  description  it  may  be  selected  from  a  collection 
of  different  solids. 

13.  Eecall  the  things  that  you  have  learned  about 
the  sphere  and  the  hemisphere. 

14:.  Draw  the  sphere  and  the  hemisphere,  and 
write  a  comparison  of  the  two  forms.  In  the  compari- 
son use  the  terms  uniformly,  radius,  radii,  diameters, 
circle  and  circumference. 

THE  HEMISPHERE  AND  THE  QUADRANT  OF  A  SPHERE. 

Make  a  sphere  of  clay.  Separate  the  sphere  into 
hemispheres.  Separate  one  of  the  hemispheres  into  two 
similar  and  equal  parts. 

Compare  the  hemisphere  with  the  quadrant,  or 
quarter  of  the  sphere. 


1.  Find  likenesses. 

2.  Find  differences. 


62  LESSON'S   IN   FORM. 

3.  What  objects  did  you  find  in  the  shape  of  a 
hemisphere  ? 

QUESTIONS— LIKENESSES. 

1.  The  hemisphere  and  the  quadrant  of  the  sphere 
are  bounded  by  what  kind  of  surfaces? 

2.  Each  has  at  least  one line. 

3.  To  what  is  the  sum  of  the  plane  surfaces  of  the 
quadrant  equal  ? 

4.  To  what  is  the  sum  of  the  two  curved  lines  of 
the  quadrant  equal? 

5.  What  is  the  name  of  the  line  in  the  hemi- 
sphere of  which  each  of  the  curved  lines  of  the  quadrant 
is  equal  to  one-half. 

DIFFERENCES. 

1.  Each  solid  is  limited  by  how  many  surfaces? 

2.  The  surfaces  of  each  are  limited  b}^  how  many 
lines  ? 

3.  What  kind  of  a  line  do  3^ou  find  in  the  quad- 
rant that  is  not  found  in  the  hemisphere? 

4.  Besides  the  straight  line,  what  else  do  you  find 
in  the  quadrant  that  is  not  found  in  the  hemisphere  ? 

5.  How  many  points  do  you  find  in  the  quadrant  ? 

6.  How  many  semi-circles  limit  the  quadrant  ? 

7.  The  curved  surface  of  the  quadrant  equals 
what  part  of  the  curved  surface  of  the  hemisphere  ? 

8.  Is  the  curved  surface  of  any  quadrant  of  a 
sphere  equal  to  one-half  the  curved  surface  of  an}^ 
hemisphere  ? 

9.  The  difference  in  direction  of  two  plane  sur- 
faces is  a  dihedral  angle. 

Find  dihedral  angles  in  the  room. 


QUADRANT  AND  THE  EIGHTH  OF  A  SPHERE.     63 

When  you  point  to  a  dihedral  angle,  say  thao  the 
difference  in  direction  —  indicating  the  directions  of  the 
two  planes  by  moving  hand  or  pointer  —  of  this  plane 
and  that  plane  is  a  dihedral  angle.     , 

Remark  :  To  show  the  direction  of  the  planes  the  pointer 
should  be  moved  in  a  line  perpendicular  to  the  intersection  of  the 
two  planes. 

10.  Does  the  quadrant  have  a  dihedral  angle  ? 
Wiiy? 

11.  How  many  things  do  j^ou  find  in  the  quad- 
rant that  are  not  found  in  the  hemisphere  ?  In  the 
hemisphere  that  are  not  found  in  the  sphere  ? 

Draw  the  hemisphere  and  the  quadrant  of  the 
sphere. 

Write  a  comparison  of  the  hemisphere  with  the 
quadrant  of  the  sphere. 

1.  Write  likenesses. 

2.  Write  differences. 

THE  QUADRANT  AND  THE  EIGHTH  OF  A  SPHERE. 

Make  a  sphere  of  clay. 

Separate  the  sphere  into  hemispheres. 

Separate  one  of  the  hemispheres  into  a  quadrant 
of  a  sphere. 

Pass  a  Avire  tlirough  the  middle  point  of  the  straight 
line  of  the  quadrant  and  separate  the  quadrant  into 
two  equal  parts. 

What  part  of  a  sphere  is  one  of  the  two  equal 
parts  of  a  quadrant  of  a  spliere  ? 

Compare  the  quadrant  of  a  sphere  with  the  eighth 
of  a  sphere. 


64 


LESSOXS   IX   FORM. 


1.  Find  likenesses. 

2.  Find  differences. 

QUESTIONS— LIKENESSES. 

1.  Each  solid  is  limited  by  what  kind  of  sur- 
faces ? 

2.  Each  solid  is  limited  by  how  many  curved 
surfaces  ? 

3.  The  surfaces  of  each  solid  are  limited  by  what 
kind  of  lines  ? 

4.  The  lines  of  each  surface  are  limited  by  what? 

5.  The  difference  in  direction  of  two  plane  sur- 
faces is  what  kind  of  an  angle? 

6.  Each  surface  has  at  least  one angle. 

7.  Is  the  difference  in  direction  of  the  two  plane 
surfaces  of  the  quadrant  of  the  sphere  equal  to  the 
difference  in  direction  of  two  of  the  plane  surfaces  of 
the  eighth  of  the  sphere  ? 

DIFFERENCES. 

1.  Each  solid  is  limited  by  how  many  plane  sur- 
faces ? 

2.  The  surfaces  of  each  solid  are  limited  by  how 
many  straight  lines  ? 

3.  The  lines  of  each  solid  are  limitea  by  how 
many  points  ? 


QUADRANT  AND  EIGHTH  OF  A  SPHERE.       65 

4.  In  the  quadrant  of  the  S})here,  two  points  are 
the  limit  of  how  many  lines? 

5.  In  the-cighth  of  the  sphere,  four  points  are  the 
limit  of  how  many  lines? 

6.  One  of  the  plane  surfaces  of  the  eighth  of  the 
sphere  is  equal  to  what  part  of  one  of  the  ])laue  sur- 
faces of  the  quadrant  of  the  sphere  ? 

7.  The  sum  of  the  plane  surfaces  of  the  eighth  of 
the  sphere  is  equal  to  what  part  of  the  sum  of  the  plane 
surfaces  of  the  quadrant  of  the  sphere  ? 

8.  One  of  the  straight  lines  of  the  eighth  of  the 
sphere  is  equal  to  what  part  of  the  straight  line  of  the 
quadrant  of  the  sphere  ? 

0.  The  straight  line  of  the  quadrant  is  equc/i  to 
v\'hat  part  of  the  sum  of  the  straight  hues  of  the  eighth 
of  the  sphere  ? 

10.  One  of  the  curved  lines  of  the  eighth  of  tlie 
s])here  is  equal  to  what  pai't  of  one  of  the  curved  lines 
of  the  quadrant  ? 

11.  The  sum  of  the  curved  lines  of  the  eighth  of 
the  sphere  is  equal  to  what  part  of  tlie  sum  of  the  curved 
lines  of  the  quadrant? 

12.  Each  solid  has  how  many  dihedral  angles? 

13.  The  difference  in  direction  of  three  oi'  inore 
plane  surfaces  is  a  solid  angle.  What  does  the  eighth 
of  a  sphere  have  that  the  quadrant  does  not  have? 

14.  Find  solid  angles  in  the  room.  Example: 
The  difference  in  direction  of  the  two  walls  and  the 
ceiling  of  the  I'oom  is  a  solid  angle. 

15.  IIow  many  solid  angles  has  a  cube?  How 
many  diliedral  angles  \ 


66  LESSOIS^S   IK   FORM. 

16.  What  is  the  difference  in  direction  of  the  three 
plane  surfaces  of  the  eighth  of  the  sphere  called  ? 

17.  The  straight  lines  of  the  eighth  of  the  sphere 
extend  in  how  many  directions? 

18.  How  many  straight  lines   are   necessary  to 
represent,  or  show,  a  difference  in  direction  ? 

19.  The  straight  lines  of  the  eighth  of  the  sphere 
form  how  many  angles  ? 

20.  How  many  pairs  of  perpendicular  lines  are 
there  in  the  eighth  of  the  sphere? 

21.  The  eighth  of  the  sphere  has  how  many  right 
angles  ? 


22.  What  did  you  find  in  the  hemisphere  that  is 
not  found  in  the  sphere  ? 

23.  What  did  you  find  in  the  quadrant  of  the 
sphere  that  is  not  found  in  the  hemisphere  ? 

24:.  What  was  found  in  the  eighth  of  the  sphere 
that  was  not  found  in  the  quadrant  of  the  sphere? 

25.  What  was  found  in  the  eighth  of  the  sphere 
that  was  not  found  in  the  sphere  ? 

Observe  and  draw  the  quadrant  of  the  sphere  and 
the  eighth  of  the  sphere.  Write  a  comparison  of  the 
quadrant  of  the  sphere  and  the  eighth  of  the  sphere. 


CUBE   AND   PYRAMID.  67 

EIGHTH  OF  A  SPHERE  AND  A  TRIANGULAR  PYRAMID. 


Make  and  compare  the  eighth  of  a  sphei^e  witli  the 
triangular  pyramid . 

1.  Find  likenesses. 

2.  Find  differences. 

3.  Draw  each  and  write  a  comparison. 

TWO-INCH  CUBE  AND  THE  TRIANGULAR  PYRAMID. 

Make  models  of  the  2-inch  cube  and  the  triangular 
pyramid. 

Suggestion  :  Draw  on  thick,  tough  cardboard  the  diagrams 
shown  below.  Cut  the  diagrams  out  of  the  cardboard  and  cut  half 
through  the  edges  of  the  surfaces  that  are  joined.  Then  fold  the 
figures  into  the  solids  to  be  compared.  Fasten  the  adjacent  surfaces 
of  the  solids  by  means  of  mucilage  or  paste. 


Observe  and  compare  the  models  you  liave  made. 


68 


LESSO"N"S   IN"   FORM. 


Draw  the  2-inch  cube  and  the  triangular  pyramid 
and  write  a  comparison  of  them. 

1.  Likenesses. 

2.  Differences. 

3.  Write  the  names  of  five  cubes. 

From  the  following  diagrams,  models  of  the  (1) 
quadrangular  prism,  (2)  triangular  prism,  (3)  hexagonal 
prism,  (4)  cylinder,  (5)  cone,  (6)  frustrum  of  a  cone,  (7) 
quadrangular  pyramid,  (8)  tetrahedron,  (9)  octahedron, 
and  (10)  icosahedron  can  be  developed. 

A  comparison  of  these  forms  will  serve  both  as 
language  exercises  and  as  an  introduction  to  the  study 
of  solid  geometry. 


^^^-^^—     \  1^^— ^--— ■ 


A 

COMPARISON    OF   FORMS. 


69 


70 


LESSONS  IN   FORM. 


POINTS,   LINES,   SURFACES    AND  SOLIDS. 

QUESTIONS. 

Suggestion:     Omit  in  the  primary  grades. 

1.  If  a  square  is  moved  so  that  it  continues  par- 
allel to  its  first  position,  and  in  one  direction,  what  will 
its  edges  generate  ?  What  will  the  four  points  of  the 
square  generate?  How  do  the  lines  generated  by  the 
four  points  compare  as  to  length  and  direction?  The 
edges  of  the  square  generate  what  kind  of  figures  ? 
Why?  Do  you  know  whether  the  parallelograms  gen- 
erated are  rhomboids  or  rectangles?     Why  not?     If 


POINTS,    LI  XLS,    SURFACES   AND   SOLIDS.  71 

the  edges  generate  rectangles,  what  kind  of  a  solid  does 
the  square  generate  ? 

2.  If  a  square  is  moved  so  that  its  edges  gene- 
rate surfaces  which  are  perpendicular  to  the  square  in 
its  first  position,  what  kind  of  a  solid  will  the  square 
generate  ? 

3.  How  far  and  in  what  manner  must  a  square 
be  moved  to  generate  a  cube  ? 

4.  How  may  a  cube  be  generated,  beginning  with 
a  point? 

5.  If  a  rectangle  be  moved  about  one  of  its 
edges  as  an  axis,  what  will  the  extremities  of  the  line 
parallel  to  this  axis  generate?  What  will  the  line  par- 
allel to  the  axis  generate  ?  AVhat  figures  will  the  lines 
perpendicular  to  the  axis  generate?  What  will  the 
rectangle  generate  ? 

6.  How  may  a  cylinder  be  generated? 

7.  IIow  may  a  cylinder  be  generated,  beginning 
with  a  point  ? 

8.  If  a  circle  be  moved  so  as  to  continue  parallel 
to  its  first  position,  and  so  that  its  center  continues  in 
a  line  perpendicular  to  the  circle  at  its  middle  point,  in 
its  first  position,  what  kind  of  a  solid  will  the  circle 
generate? 

9.  IIow  must  a  circle  be  moved  to  generate  a 
cylinder?  On  how  many  of  the  surfaces  of  a  cyliiuk'r 
can  straiglit  lines  be  drawn? 

10.  If  a  right  triangle  be  turned  on  one  of  its 
perpendicular  edges  as  an  axis,  Avhat  will  it  generate? 
What  will  the  line  perpendicular  to  the  axis  generate? 
What  line  of  the  circle  will  one  of  the  extremities  of 


72  LESSONS   IN"   FORM. 

the  perpendicular  generate?     The  other  extremity  of 
the  perpendicuhir  will  be  what  point  of  the  circle? 

11.  If  a  circle  be  turned  on  one  of  its  diameters 
as  an  axis,  what  will  it  generate  ?  What  lines  of  the 
circle  and  sphere  are  equal  ? 

12.  How  may  a  sphere  be  generated  ? 

13.  Can  a  hemisphere  be  generated  by  moving  a 
circle  in  any  manner  ? 

14.  How  must  a  serai-circle  be  moved  to  generate 
a  sphere  ? 

15.  If  a  semi-circle  makes  half  a  revolution  on  its 
straight  edge  as  an  axis,  what  will  it  generate? 

1.  A  cube  is  a  solid,  so  is  any  portion  of  space 
limited  by  a  surface  or  surfaces. 

2.  How  many  dimensions  has  a  solid  ? 

3.  How  many  dimensions  has  a  part  of  a  solid? 

4.  IIoAV  many  dimensions  has  a  surface  ? 

5.  Is  a  surface  a  part  of  a  solid?     Why  not  ? 

6.  Does  a  surface  have  thickness  ?  Would  several 
surfaces  have  tliickness  ?  Would  several  surfaces 
make  a  solid  ?     Wliy  not  ? 


WI-IY  ELEMENTARY  FOKM  LESSONS 

SHOULD  PBECEDE  THE  DIRECT 

STUDY  OF  NUMBER, 


The  Nature  of  Number — How  Ideas  of  Number 
ARE  Gained. —  Mathematics  is  a  knowledge  of  the  lim- 
itations of  quantity.  These  limitations  are  of  two 
kinds:  fonn  JimiLations  and  number  limitations. 
Number  is  but  an  abstraction.  It  is  not  the  quantity 
nor  a  quahty  of  the  quantity.^  That  numbei'  is  an 
idea  whicli  accompanies  the  sense  of  sight,  touch,  or 
hearing,  and  is  not  an  impression  of  the  sense  itself, 
was  held  by  Aristotle,  Locke,  Hamilton  and  others. 
The  child  can  not  think  or  compare  numbers  independ- 
ent of  the  quantities  wliich  they  limit.  If  we  try  to 
think  one,  or  of  one,  we  can  not  do  it,  for  the  oneness 
does  not  exist  in  the  adjective,  but  in  that  in  which  the 

adjective  limits.  A  difference  in  the  force  of  the 
adjective  good  as  compared  witli  another  adjective 
good,  can  not  be  realized  in  thought,  for  the  dif- 
ference exists  in  the  things  w^hich  these  adjectives 
describe ;  as,  a  good  boy,  a  good  apple.  The  difference 
in  the  qualities  of  these  adjectives  must  be  sought  in  an 
examination   of   the  things   limited,    and   not    in   the 

*  "  Number  in  the  abstract  is,  of  course,  a  merely  intellectual  concept, 
as  Aristotle  once  and  a^ain  notices."— Siii  Wm.  Hamilton. 

"It  is  evident  that  number,  far  from  bein<?  a  quality  of  matter,  is  only 
an  abstract  notion,— the  work  of  tlie  iutcliect  and  not  of  the  sense."— 

ItOYiUi-COLLARD. 

73 


74  LESSONS   IK   FOEM. 

adjectives  apart  from  the  things.  We  must  proceed  in 
the  same  way  if  we  would  learn  to  use  numeral  adjec- 
tives, or  numbers,  intelligently.  Until  a  pupil  has  a 
mental  picture  of  an  inch  cube  and  of  a  3-incli  cube  he 
cannot  see  for  himself  what  part  one  is  of  the  other ;  if 
he  cannot  see  mentally  the  relation  of  one  edge  of  tlie 
inch  cube  to  one  edge  of  the  3-inch  cube,  or  of  1  edge 
of  the  1-27  to  1  edge  of  the  27-27,  he  cannot  think  the 
cube  root  of  1-27.  Nor  can  he  compare  1-2  with  1-3 
unless  he  can  see  their  relation  to  each  other  throiioh 
seeing  their  relation  to  6-6,  or  1 ;  and  to  see,  or  image, 
6-6,  is  to  see,  or  image,  the  6  equal, parts  of  some 
quantity. 

The  Development  of  the  Image-Forming  Power 
THE  Basis  of  Mathematical  In^/estigation. — All  reason- 
ing in  arithmetic  is  based  on  seeing  conditions,  and 
ability  to  see  conditions  is  based  on  ability  to  think  the 
relations  of  quantities,  and  not  the  relations  of  numbers, 
and  to  see  the  relations  of  quantities,  the  quantities 
(their  correspondents,  of  course)  must  be  in  the  mind 
for  examination.  The  necessary  antecedent,  then  of 
the  formal  teaching  of  mathematics  is  the  training  of 
the  imagination;"^  such  a  training  as  will  give  to  the 
child  clearly  defined  concepts  of  things  wliich  he  can 
recall,  analyze,  and  compare  at  will.  Without  clear 
concepts  he  has  no  data  U2:)on  which  to  base  his  reason- 

*  "  If  the  imagination  have  not  a  sound  basis  in  habits  of  accurate 
observation,  it  degenerates  Into  fancy  ;  a  term  which,  though  originally  it 
was  considered  to  mean  the  same  thing  as  imagination,  is  now  used  to  denote 
a  well-founded  difference.  Fancy  represents  the  productive  or  creative 
power  of  imagination  working  without  that  due  resti-aint  of  law  which  is 
imposed  upon  its  operation  by  habits  of  accurate  observation,  and  without 
that  proper  and  sufficient  material  of  facts  which  such  observation 
furnishes. ' '— M  audsle  y, 


FORM    LESSON'S   TO    PRECEDE   NUMBER   STUDY.  75 

ing  and  conclusions.  A  pupil's  real  progress  in  num- 
ber is  measured  by  his  power  to  think  of  things  inde- 
pendent of  their  concrete  manifestation..  To  our  losing 
sight  of  these  facts,  and  attempting  to  teach  number 
before  a  proper  basis  has  been  laid  by  cultivating  the 
image-forming  power,  may  be  attributed  the  disHke  to 
the  subject,  and  the  almost  incredible  inability  on  the 
part  of  a  majority  of  pupils  to  solve  simple  problems. 

Why  Some  Ideas  of  Form,  Color,  Direction,  Posi- 
tion, Size,  Etc.,  Should  be  Taught  Before  Beginning 
THE  Formal  Study  of  jS^umber. — As  clear  concepts  result 
from  distinct  precepts,  and  these,  in  turn,  from  repeated 
observations,  it  is  evident  that  the  natural  approach  to 
number  teaching  is  through  things  upon  which  Ave  can 
fix  the  attention  of  the  children. 

It  is  true  that  objects  are  used  in  teaching  arithme- 
tic in  most  of  our  primary  schools ;  but  is  it  true  that 
they  are  usually  so  selected,  and  so  used,  as  to  furnish 
an  objective  basis  for  the  mind  in  its  operations  ?  Hand- 
ling objects  is  of  little  value  if  nothing  corresponding  to 
them  comes  into  the  mind. 

Little  power  is  gained  to  hold  an  apple  m  the  mind 
by  saying  2  apples  and  2  apples  are  4  apples.  The 
power  to  image  an  apple  depends  on  the  ideas  you  have 
gained  of  it,  either  by  direct  study  or  incidentally.  It 
is  self-evident  that  an  object  of  which  you  know  noth- 
ing except  that  there  is  one  or  two  is  not  in  the  mind, 
and  cannot  be  thought  of,  and,  that  the  more  ideas  you 
have  of  an  object,  the  more  perfect  is  the  concept,  and, 
by  the  la\vs  of  association,  the  more  easily  it  is  held  in 
the  mind.  If  the  likenesses  and  differences  of  two 
apples  were  considered,  their  color,  form,  size,  weight, 


76  LESSONS   IN"   FORM. 

etc.,  the  effect  of  such  a  lesson  on  future  mathematical 
investigation  would  be  far  greater  than  if  attention  were 
turned  first  and  solely  to  the  number. 

^  The  number  relations  of  4  apples  are  just  the  same 
as  the  number  relations  of  4  of  anything  else.  The  dif- 
ferences exist  in  the  things.  Then  to  pass  from  4  of 
one  thing  to  4  of  another  is  to  pass  to  exactly  the  same 
thing,  if  ihe  number  only  is  considered.  These  repeti- 
tions soon  become  wearisome;  the  stimulus  to  thought 
groAvs  less  and  less  with  each  repetition,  and  the  inter- 
est dies. 

Saying  that  5  squares  and  2  squares  are  7  squares 
does  not  lead  to  a  close  observation  of  the  squares. 
Children  do  not  see  the  four  equal  lines,  the  opposite 
lines,  the  two  pairs  of  parallel  lines,  the  four  pairs  ot 
adjacent  lines,  the  four  pairs  of  perpendicular  lines,  the 
four  right  angles,  the  opposite  points  and  opposite 
angles,  and  the  convergence  and  the  divergence  of  the 
lines.  These  things  are  overlooked  in  the  consideration 
of  the  how  many  onh\  As  it  is  with  the  squares  so  is 
it  with  other  objects  used  for  number  exercises.  Atten- 
tion must  be  given  to  something  else  in  the  object  if 
the  exercise  is  to  foster  habits  of  investigation  which 
will  lead  to  a  living  apprehension  of  the  relations  of 
quantitv. 

I  do  not  here  urge  the  study  of  the  properties  of 
the  apple,  the  square,  etc.,  for  the  sake  of  a  knowledge 
of  these  things,  but  because  I  believe  it  to  be  the  best 
means  of  training  the  observing  powers  and  the  imag- 
ination, and  therefore  the  most  economical  basis  for  the 
study  of  arithmetic.  Things  are  held  in  the  mind  by 
their  form  and  not  by  their  number,  and  a  pupil  who 


FORM    LESSON'S   TO    PRECEDE   I^UMBER   STUDY.  77 

studies  form  and  natural  science  one  3^ear,  letting  num- 
ber be  incidental,  and  then  begins  the  direct  study  of 
number,  witli  these  studies  in  a  parallel  course,  will 
know  much  more  of  number  in  three  years  than  one 
who  studies  number  alone  from  the  beginning. 

The  mere  NUMBERixa  of  objects  does  not  build 
THEM  INTO  THE  MIND ;  if  it  did,  wc  could  learn  form  and 
natural  science  by  merely  numbering  forms,  flowers, 
etc.  Hence,  in  a  typical  primary  school  where  balls, 
shoe-pegs,  apples,  blocks  and  toys  are  used  in  element- 
tary  number  lessons,  Ave  find  pupils  unable  to  think  in 
number.  They  can  count  and  repeat  tables,  but  have 
little  power  to  investigate  new  conditions — to  verify 
or  disprove  the  conclusions  of  others.  They  have  not 
gained  objects  of  thought  which  they  can  compare. 
Saying  that  5  beans  and  1  bean  are  6  beans,  that  6 
beans  less  3  beans  are  3  beans,  etc.,  does  not  awaken 
an  interest  in  the  relations  of  numbers,  but  usually  de- 
generates into  mere  mechanical  drill,  through  which 
expressions  are  memorized.  The  repetitions  necessary 
to  nicike  ])ermanent  ^possessions  of  the  things  pei'ceived  are 
not  repetitions  of  words  hut  of  ideas  ^  and  to  get  the  thought 
again  and  again  hfore  the  jpiipil  is  only  possible  hy 
arousing  tJie  mind  to  activity. 

The  Relations  of  I^umber  Seen  through  Seeing 
THE  Relations  of  Quantity. — The  common  practice  is 
to  give  all  the  attention  to  the  number  and  none  to  the 
quantity.  Tlie  converse  of  this  is  not  recommended, 
but  that  the  study  of  quantity  should  be  so  pui'sued  as 
to  create  a  demand  for  limitation  by  nnniber.  Just  as 
we  teach  a  word  when  a  child  has  an  idea  for  which  it 
needs  expression,  so  should  we  teach  number  when  its 


78  LESSORS   IN"   FORM. 

help  is  needed  to  definitely  limit  quantities.  A  child 
discovers  a  difference  in  the  length  of  two  Ihies.  There 
is  then  a  demand  for  units  of  measure  ^ — units  of  quan- 
tity—  and  for  numeral  adjectives  that  the  difference 
may  be  clearly  defined  in  thought,  and  exactly  ex- 
pressed. 

FoKM  Better  Adapted  than  any  Other  Subject 
FOR  Beginning  Systematic  Training  of  Perception  and 
Imagination;  Therefore,  the  Best  Basis  for  Number 
Study. —  Pestalozzi,  Froebel,  and  tlie  kindergartners 
generally,  recognize  the  value  of  form  both  as  a  direct 
means  of  training  the  perceptive  faculties  and  as  a  prep- 
aration for  the  study  of  mathematics. 

Geometrical  forms  are  simple  in  outline  and  pos- 
sess distinctive,  clearly -marked  features  Avhich  the  child- 
ish mind  can  readily  grasp.  These  features  are  con- 
stantly repeated,  but  in  a  variety  of  forms,  so  that  the 
repititions  do  not  weary.  The  relations  of  the  parts 
of  the  divided  solids  to  the  whole  and  to  each  other  are 
exact,  so  that  both  through  analysis  and  construction 
facts  of  form  and  numbers  may  be  discovered  and  fixed. 
Children  should  begin  with  the  simple  and  exact,  and 
pass  to  the  complex  and  indefinite.^ 

Form  furnishes  a  better  means  for  the  beginning 
lessons  in  observation  than  does  natural  science.  Many 
objects  in  natural  science  are  irregular  — the  measure- 
ments of  their  parts  not  exact.  Children  can  not  ex- 
press definitely  the  likenesses  and  differences  they  dis- 
cover ;  hence,  their  thought  to  a  certain  extent  is  in- 

*  "  The  possibilities  of  the  object  must  not  be  too  varied,  and  it  must 
be  sugg^estive  through  its  limitations.  The  young  mind  may  be  as  easily 
crushed  by  excess  as  by  defect."— Susan  E.  Blow. 


FORM   LESSONS   TO    PRECEDE    NUMBER   STUDY.  79 

distinct.  A  description  in  which  only  such  terms  are 
used  as  long,  slender,  short,  etc.,  lacks  in  precision. f 
Xo  other  study  is  so  well  adapted  for  close  com- 
parison in  the  lower  grades  as  the  study  of  form, 
because  in  no  other  are  the  likenesses  and  differences  so 
clearly  marked ;  hence  this  subject  best  trains  the  dis- 
criminating power  wdiich  leads  to  classification  and 
generalization, 

Without  elementary  ideas  of  form  neither  geog- 
raphy, drawing  or  natural  science,  can  be  properly 
taught.  Progress  in  other  subjects  w^ill  be  greatly 
facilitated  by  understanding  the  language  of  form. 

There  would  be  but  little  difficult}^  in  teaching 
mensuration,  in  arithmetic,  if  pupils  were  able  to  think 
of  the  forms  with  which  they  deal. 

Scientific  geometry  cannot  be  intelligently  taught 
without  elementary  ideas  of  the  subject  gained  through 
the  study  of  form  in  the  concrete.  These  ideas  should 
be  gained  in  the  perceptive  stage  of  education,  and  not 
when  the  student  ought  to  be  exercising  his  reasoning 
powers.  W.  W.  S. 

"  The  infant  begins  to  examine  formsfrom  the  commencement 
of  his  existence;  for  without  this  knowledge  it  isdoubtful  if  he  could 
distinguish  one  object  from  another,  or  even  be  aware  of  an  external 
world .  Gradually  he  begins  to  know  objects  apart  and  to  recognize 
them,  and  in  time  discerns  resemblances  which  cause  him  to  classify 
them.  A  vast  amount  of  time  and  labor  is  spent  by  every  child  in 
the  investigations  during  the  first  ten  years  f  f  h's  life  ;  but  not  more 
than  their  importance  requires.  Every  child  is  therefore  in  some 
degree  a  self-taught  geometer.  Can  it  tlicn  be  said  that  form  is  not 
suited  for  early  education?"     .     . 

+  "  It  is  in  mathematical  science  alone  that  words  arc  the  signs  of  ex- 
act and  clearly  defined  ideas.  It  is  here  alone  that  wo  can  pee,  as  it  were, 
the  very  thoiifjrhts  through  the  transparent  words  by  which  they  are  ex- 
pressed."—Da  vies. 


80  LESSONS   IN   FORM. 

"  Half  a  dozen  simple  poiuts  investigated  and  discovered  by  the 
pupil  will  be  of  more  value  than  a  book  full  of  geometry,  to  vrbich 
he  merely  gives  a  cold  assent." 

Horace  Grakt. 

"Geometrical  facts  and  conceptions  are  easier  to  a  child  tban 
those  of  arithmetic." 

Ex-President  Hill,  of  Harvard  College 

"Mathematics  is  the  only  exact  science;  if  tbe  premises  are 
correct  the  conclusions  must  be.  To  form  a  strong  effectual  habit 
of  seeing  and  thinking  of  things  just  as  they  are  and  in  their  exact 
relations,  is  the  province  of  mathematics."     .     .     . 

"  Ideas  grow  slowly.  It  takes«a  long  time,  with  many  acts  of 
perception,  to  fix  an  idea  clearly  in  the  mind.  It  is  of  immeiise 
importance  that  these  ideas  come  into  the  mind  so  distinctly  that 
they  can  be  used  in  thinking."  Col.  F.  W.  Parker. 

"The elements  of  geometry  are  much  easier  to  learn,  find  are 
of  more  value  when  .earned,  than  advanced  ariihmctic;  and  if  a  boy 
is  to  leave  school  with^merely  a  grammar-school  education,  he  would 
be  better  prepared  for  the  active  duties  of  life  with  a  little  arithmetic 
and  some  geometry,  than  with  more  arithmetic  and  no  geometry." 

Prof.  Marks. 

"That  pupil  is  fortunate  who  has  really  had  good  object- 
lessons  in  form  at  and  'rom  an  early  age.  ...  In  all  his  teach- 
ing he  must  not  forget  that  the  end  in  view  is  to  produce  images  of 
the  geometrical  figures  in  the  minds  of  his  pupils;  so  that  he  and  they 
will  be  looking  mentally  at  the  same  or  similar  objects,  and  that 
neither  will  be  lost  among  the  abstract  words." 

T.  H.  Safford,  Professor  of  Astronomy  in  Williams  College. 

"  However  excellent  arithmetic  may  be  as  an  instrument  for 
strengthening  the  intellectual  powers,  geometry  is  far  more  so ;  for 
as  it  is  easier  to  see  the  relation  of  surface  to  surface  and  of  line  to 
line  than  of  one  number  to  another,  so  it  is  easier  to  induce  a  habit 
of  '^easoningby  meansof  geometry  than  it  is  by  means  of  arithmetic." 
Wm.  George  Spencer,  the  father  of  Herbert  Spencer. 

"  When  the  understanding  is  once  stored  with  these  simple 
ideas,  it  has  the  power  to  repeat,  compare,  and  unite  them,  even  to  an 


FORM   LESSON'S  TO    PRECEDE   N-UMBER   STUDY.  81 

almost  infinite  variety,  and  so  can  make  at  pleasure  new,  complex 
ideas.  But  it  is  not  in  the  power  of  the  most  exalted  wit,  or  enlarged 
understanding,  by  any  quickness  or  variety  of  thought,  to  invent  or 
frame  one  new  simple  idea  in  the  mind  not  taken  in  by  the  ways 
aforementioned."  Locke. 

"  Instruction  must  begin  with  actual  inspection,  not  with  verbal 
descriptions  of  things.  From  such  inspection  it  is  that  certain 
knowledge  comes.  What  is  actually  seen  remains  faster  in  the 
memory  than  description  or  enumeration  a  hundred  times  as  often 
repeated."  Comenius. 

"  Observation  is  the  absolute  basis -of  all  knowledge.  The  first 
object  then,  in  education,  must  be  to  lead  the  child  to  observe  with 
accuracy;  the  second,  to  express,  with  correctness,  the  results  of  his 
observation."  Pestalozzi. 

"  If  we  consider  it,  says  Herbert  Spencer,  "  we  shall  find  that 
eidiaustive  observation  is  an  element  of  all  great  success." 

"The  education  of  the  senses  neglected,  all  after-education 
partakes  of  a  drowsiness,  a  haziness,  an  insufilciency,  which  it  is 
impossible  to  cure."  Bacon. 

Please  read  what  Agassiz  says  of  the  value  of  comparisons.     See 
page  20 . 


/" 


7/ 


^  ^^* 


^^^wj^*^m^mm^iemmm^^ 


% 


*♦ 


'M 


**;j^ 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  50  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $1.00  ON  THE  SEVENTH  DAY 
OVERDUE. 


4#: 


^^?*i 


i>..  #  w 


>!V' 


«   I 


^fJ^»: 


'■■,;  % 


»;^^:Y»±\»xh^'i:<**\v^ 


f  . .♦r^-;?*;;^' •,  ♦  ,«>.-.'n-,^vv.». 


Mt 


*     ♦ 


>\\^ 


^.^^|^^^ 


>\^\\«\\a: 


